Search: a072102 -id:a072102
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A001597
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Perfect powers: m^k where m > 0 and k >= 2.
(Formerly M3326 N1336)
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+10
568
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1, 4, 8, 9, 16, 25, 27, 32, 36, 49, 64, 81, 100, 121, 125, 128, 144, 169, 196, 216, 225, 243, 256, 289, 324, 343, 361, 400, 441, 484, 512, 529, 576, 625, 676, 729, 784, 841, 900, 961, 1000, 1024, 1089, 1156, 1225, 1296, 1331, 1369, 1444, 1521, 1600, 1681, 1728, 1764
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graph;
refs;
listen;
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internal format)
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OFFSET
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1,2
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COMMENTS
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a(1) = 1, for n >= 2: a(n) = numbers m such that sum of perfect divisors of x = m has no solution. Perfect divisor of n is divisor d such that d^k = n for some k >= 1. a(n) for n >= 2 is complement of A175082. - Jaroslav Krizek, Jan 24 2010
Catalan's conjecture (now a theorem) is that 1 occurs just once as a difference, between 8 and 9.
For a proof of Catalan's conjecture, see the paper by Metsänkylä. - L. Edson Jeffery, Nov 29 2013
m^k is the largest number n such that (n^k-m)/(n-m) is an integer (for k > 1 and m > 1). - Derek Orr, May 22 2014
a(n) is asymptotic to n^2, since the density of cubes and higher powers among the squares and higher powers is 0. E.g.,
a(10^1) = 49 (49% of 10^2),
a(10^2) = 6400 (64% of 10^4),
a(10^3) = 804357 (80.4% of 10^6),
a(10^4) = 90706576 (90.7% of 10^8),
a(10^n) ~ 10^(2n) - o(10^(2n)). (End)
a(10^n): 1, 49, 6400, 804357, 90706576, 9565035601, 979846576384, 99066667994176, 9956760243243489, ... . - Robert G. Wilson v, Aug 15 2014
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REFERENCES
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R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 1990, p. 66.
René Schoof, Catalan's Conjecture, Springer-Verlag, 2008, p. 1.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
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H. W. Gould, Problem H-170, Fib. Quart., 8 (1970), p. 268, problem H-170.
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FORMULA
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Goldbach showed that Sum_{n >= 2} 1/(a(n)-1) = 1.
Formulas from postings to the Number Theory List by various authors, 2002:
Sum_{i >= 2} Sum_{j >= 2} 1/i^j = 1;
Sum_{k >= 2} 1/(a(k)+1) = Pi^2 / 3 - 5/2;
Sum_{k >= 2} 1/a(k) = Sum_{n >= 2} mu(n)(1- zeta(n)) approx = 0.87446436840494... See A072102.
For asymptotics see Newman.
a(n) = n^2 - 2*n^(5/3) - 2*n^(7/5) + (13/3)*n^(4/3) - 2*n^(9/7) + 2*n^(6/5) - 2*n^(13/11) + o(n^(13/11)) (Jakimczuk, 2012). - Amiram Eldar, Jun 30 2023
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MAPLE
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isA001597 := proc(n)
local e ;
e := seq(op(2, p), p=ifactors(n)[2]) ;
return ( igcd(e) >=2 or n =1 ) ;
end proc:
option remember;
local a;
if n = 1 then
1;
else
for a from procname(n-1)+1 do
if isA001597(a) then
return a ;
end if;
end do;
end if;
end proc:
N:= 10000: # to get all entries <= N
sort({1, seq(seq(a^b, b = 2 .. floor(log[a](N))), a = 2 .. floor(sqrt(N)))}); # Robert FERREOL, Jul 18 2023
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MATHEMATICA
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min = 0; max = 10^4; Union@ Flatten@ Table[ n^expo, {expo, Prime@ Range@ PrimePi@ Log2@ max}, {n, Floor[1 + min^(1/expo)], max^(1/expo)}] (* T. D. Noe, Apr 18 2011; slightly modified by Robert G. Wilson v, Aug 11 2014 *)
perfectPowerQ[n_] := n == 1 || GCD @@ FactorInteger[n][[All, 2]] > 1; Select[Range@ 1765, perfectPowerQ] (* Ant King, Jun 29 2013; slightly modified by Robert G. Wilson v, Aug 11 2014 *)
nextPerfectPower[n_] := If[n == 1, 4, Min@ Table[ (Floor[n^(1/k)] + 1)^k, {k, 2, 1 + Floor@ Log2@ n}]]; NestList[ nextPerfectPower, 1, 55] (* Robert G. Wilson v, Aug 11 2014 *)
Join[{1}, Select[Range[2000], GCD@@FactorInteger[#][[All, 2]]>1&]] (* Harvey P. Dale, Apr 30 2018 *)
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PROG
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(Magma) [1] cat [n : n in [2..1000] | IsPower(n) ];
(PARI) {a(n) = local(m, c); if( n<2, n==1, c=1; m=1; while( c<n, m++; if( ispower(m), c++)); m)} /* Michael Somos, Aug 05 2009 */
(PARI) list(lim)=my(v=List(vector(sqrtint(lim\=1), n, n^2))); for(e=3, logint(lim, 2), for(n=2, sqrtnint(lim, e), listput(v, n^e))); Set(v) \\ Charles R Greathouse IV, Dec 10 2019
(Sage)
return [k for k in (1..n) if k.is_perfect_power()]
(Haskell)
import Data.Map (singleton, findMin, deleteMin, insert)
a001597 n = a001597_list !! (n-1)
(a001597_list, a025478_list, a025479_list) =
unzip3 $ (1, 1, 2) : f 9 (3, 2) (singleton 4 (2, 2)) where
f zz (bz, ez) m
| xx < zz = (xx, bx, ex) :
f zz (bz, ez+1) (insert (bx*xx) (bx, ex+1) $ deleteMin m)
| xx > zz = (zz, bz, 2) :
f (zz+2*bz+1) (bz+1, 2) (insert (bz*zz) (bz, 3) m)
| otherwise = f (zz+2*bz+1) (bz+1, 2) m
where (xx, (bx, ex)) = findMin m -- bx ^ ex == xx
(Python)
from sympy import perfect_power
def ok(n): return n==1 or perfect_power(n)
(Python)
import sympy
def __init__(self) :
self.a = [1]
def at(self, n):
if n <= len(self.a):
return self.a[n-1]
else:
cand = self.at(n-1)+1
while sympy.perfect_power(cand) == False:
cand += 1
self.a.append(cand)
return cand
for n in range(1, 20):
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CROSSREFS
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Cf. A023055, A023057, A025478, A070428, A072102, A074981, A076404, A239728, A239870, A097054, A089579, A089580.
There are four different sequences which may legitimately be called "prime powers": A000961 (p^k, k >= 0), A246655 (p^k, k >= 1), A246547 (p^k, k >= 2), A025475 (p^k, k=0 and k >= 2), and which are sometimes confused with the present sequence.
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KEYWORD
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nonn,easy,nice
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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A136141
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Decimal expansion of Sum_{p prime} 1/(p*(p-1)).
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+10
36
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7, 7, 3, 1, 5, 6, 6, 6, 9, 0, 4, 9, 7, 9, 5, 1, 2, 7, 8, 6, 4, 3, 6, 7, 4, 5, 9, 8, 5, 5, 9, 4, 2, 3, 9, 5, 6, 1, 8, 7, 4, 1, 3, 3, 6, 0, 8, 3, 1, 8, 6, 0, 4, 8, 3, 1, 1, 0, 0, 6, 0, 6, 7, 3, 5, 6, 7, 0, 9, 0, 2, 8, 4, 8, 9, 2, 3, 3, 3, 9, 7, 8, 3, 3, 7, 9, 8, 7, 5, 8, 8, 2, 3, 3, 2, 0, 8, 1, 8, 3, 2, 8, 9
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OFFSET
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0,1
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COMMENTS
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Excess of prime factors with multiplicity over distinct prime factors for random (large) integers. - Charles R Greathouse IV, Sep 06 2011
Decimal expansion of the infinite sum of the reciprocals of the prime powers which are not prime (A246547). - Robert G. Wilson v, May 13 2019
See the second 'Applications' example under the Mathematica help file for the function PrimePowerQ. - Robert G. Wilson v, May 13 2019
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REFERENCES
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Henri Cohen, Number Theory, Volume II: Analytic and Modern Tools, GTM Vol. 240, Springer, 2007; see pp. 208-209.
Steven R. Finch, Mathematical Constants, Cambridge Univ. Press, 2003, Meissel-Mertens constants, p. 94.
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LINKS
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FORMULA
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Equals 2 * Sum_{k>=2} pi(k)/(k^3-k), where pi(k) = A000720(k) (Shamos, 2011, p. 8). - Amiram Eldar, Mar 12 2024
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EXAMPLE
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Equals 1/2 + 1/(3*2) + 1/(5*4) + 1/(7*6) + ...
= 0.7731566690497951278643674598559423956187413360831860483110060673567...
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MATHEMATICA
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digits = 103; sp = NSum[PrimeZetaP[n], {n, 2, Infinity}, WorkingPrecision -> digits + 10, NSumTerms -> 2*digits]; RealDigits[sp, 10, digits] // First (* Jean-François Alcover, Sep 02 2015 *)
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PROG
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(PARI) W(x)=solve(y=log(x)/2, max(1, log(x)), y*exp(y)-x)
eps()=2. >> (32*ceil(default(realprecision)/9.63))
primezeta(s)=my(t=s*log(2), iter=W(t/eps())\t); sum(k=1, iter, moebius(k)/k*log(abs(zeta(k*s))))
a(lim, e)={ \\ choose parameters to maximize speed and precision
my(x, y=exp(W(lim)-.5));
x=lim^e*(e*log(y))^e*(y*log(y))^-e*incgam(-e, e*log(y));
forprime(p=2, lim, x+=1/((p*1.)^e*(p-1)));
x+sum(n=2, e, primezeta(n))
(PARI) sumeulerrat(1/(p*(p-1))) \\ Amiram Eldar, Mar 18 2021
(Magma) R := RealField(105);
c := &+[R|(EulerPhi(n)-MoebiusMu(n))/n*Log(ZetaFunction(R, n)):n in[2..360]];
Reverse(IntegerToSequence(Floor(c*10^103))); // Jason Kimberley, Jan 12 2017
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CROSSREFS
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KEYWORD
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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A091050
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Number of divisors of n that are perfect powers.
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+10
25
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1, 1, 1, 2, 1, 1, 1, 3, 2, 1, 1, 2, 1, 1, 1, 4, 1, 2, 1, 2, 1, 1, 1, 3, 2, 1, 3, 2, 1, 1, 1, 5, 1, 1, 1, 4, 1, 1, 1, 3, 1, 1, 1, 2, 2, 1, 1, 4, 2, 2, 1, 2, 1, 3, 1, 3, 1, 1, 1, 2, 1, 1, 2, 6, 1, 1, 1, 2, 1, 1, 1, 5, 1, 1, 2, 2, 1, 1, 1, 4, 4, 1, 1, 2, 1, 1, 1, 3, 1, 2, 1, 2, 1, 1, 1, 5, 1, 2, 2, 4, 1, 1
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OFFSET
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1,4
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COMMENTS
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LINKS
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FORMULA
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G.f.: Sum_{k=i^j, i>=1, j>=2, excluding duplicates} x^k/(1 - x^k). - Ilya Gutkovskiy, Mar 20 2017
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 1 + A072102 = 1.874464... . - Amiram Eldar, Dec 31 2023
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EXAMPLE
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Divisors of n=108: {1,2,3,4,6,9,12,18,27,36,54,108},
a(108) = #{1^2, 2^2, 3^2, 3^3, 6^2} = 5.
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MATHEMATICA
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ppQ[n_] := GCD @@ Last /@ FactorInteger@ n > 1; ppQ[1] = True; f[n_] := Length@ Select[ Divisors@ n, ppQ]; Array[f, 105] (* Robert G. Wilson v, Dec 12 2012 *)
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PROG
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(Haskell)
a091050 = sum . map a075802 . a027750_row
(PARI) a(n) = 1+ sumdiv(n, d, ispower(d)>1); \\ Michel Marcus, Sep 21 2014
(PARI) a(n)={my(f=factor(n)[, 2]); 1 + if(#f, sum(k=2, vecmax(f), moebius(k)*(1 - prod(i=1, #f, 1 + f[i]\k))))} \\ Andrew Howroyd, Aug 30 2020
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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A052486
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Achilles numbers - powerful but imperfect: if n = Product(p_i^e_i) then all e_i > 1 (i.e., powerful), but the highest common factor of the e_i is 1, i.e., not a perfect power.
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+10
20
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72, 108, 200, 288, 392, 432, 500, 648, 675, 800, 864, 968, 972, 1125, 1152, 1323, 1352, 1372, 1568, 1800, 1944, 2000, 2312, 2592, 2700, 2888, 3087, 3200, 3267, 3456, 3528, 3872, 3888, 4000, 4232, 4500, 4563, 4608, 5000, 5292, 5324, 5400, 5408, 5488, 6075
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OFFSET
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1,1
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COMMENTS
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Number of terms < 10^n: 0, 1, 13, 60, 252, 916, 3158, 10553, 34561, 111891, 359340, 1148195, 3656246, 11616582, 36851965, ..., A118896(n) - A070428(n). - Robert G. Wilson v, Aug 11 2014
a(n) = (s(n))^2 * f(n), s(n) > 1, f(n) > 1, where s(n) is not a power of f(n), and f(n) is squarefree and gcd(s(n), f(n)) = f(n). - Daniel Forgues, Aug 11 2015
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LINKS
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FORMULA
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Sum_{n>=1} 1/a(n) = zeta(2)*zeta(3)/zeta(6) - Sum_{k>=2} mu(k)*(1-zeta(k)) - 1 = A082695 - A072102 - 1 = 0.06913206841581433836... - Amiram Eldar, Oct 14 2020
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EXAMPLE
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a(3)=200 because 200=2^3*5^2, both 3 and 2 are greater than 1, and the highest common factor of 3 and 2 is 1.
Factorizations of a(1) to a(20):
72 = 2^3 3^2, 108 = 2^2 3^3, 200 = 2^3 5^2, 288 = 2^5 3^2,
392 = 2^3 7^2, 432 = 2^4 3^3, 500 = 2^2 5^3, 648 = 2^3 3^4,
675 = 3^3 5^2, 800 = 2^5 5^2, 864 = 2^5 3^3, 968 = 2^3 11^2,
972 = 2^2 3^5, 1125 = 3^2 5^3, 1152 = 2^7 3^2, 1323 = 3^3 7^2,
1352 = 2^3 13^2, 1372 = 2^2 7^3, 1568 = 2^5 7^2, 1800 = 2^3 3^2 5^2.
Examples for a(n) = (s(n))^2 * f(n): (see above comment)
s(n) = 6, 6, 10, 12, 14, 12, 10, 18, 15, 20, 12, 22, 18, 15, 24, 21,
f(n) = 2, 3, 2, 2, 2, 3, 5, 2, 3, 2, 6, 2, 3, 5, 2, 3,
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MAPLE
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filter:= proc(n) local E; E:= map(t->t[2], ifactors(n)[2]); min(E)>1 and igcd(op(E))=1 end proc:
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MATHEMATICA
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achillesQ[n_] := Block[{ls = Last /@ FactorInteger@n}, Min@ ls > 1 == GCD @@ ls]; Select[ Range@ 5500, achillesQ@# &] (* Robert G. Wilson v, Jun 10 2010 *)
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PROG
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(Python)
from math import gcd
from itertools import count, islice
from sympy import factorint
def A052486_gen(startvalue=1): # generator of terms >= startvalue
return (n for n in count(max(startvalue, 1)) if (lambda x: all(e > 1 for e in x) and gcd(*x) == 1)(factorint(n).values()))
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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Example edited by Mac Coombe (mac.coombe(AT)gmail.com), Sep 18 2010
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STATUS
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approved
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A340588
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Squares of perfect powers.
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+10
3
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1, 16, 64, 81, 256, 625, 729, 1024, 1296, 2401, 4096, 6561, 10000, 14641, 15625, 16384, 20736, 28561, 38416, 46656, 50625, 59049, 65536, 83521, 104976, 117649, 130321, 160000, 194481, 234256, 262144, 279841, 331776, 390625, 456976, 531441, 614656, 707281, 810000, 923521, 1000000
(list;
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OFFSET
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1,2
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LINKS
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FORMULA
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Sum_{k>1} 1/(a(k) - 1) = 7/4 - Pi^2/6 = 7/4 - zeta(2).
Sum_{k>1} 1/a(k) = Sum_{k>=2} mu(k)*(1-zeta(2*k)).
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MATHEMATICA
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Join[{1}, (Select[Range[2000], GCD @@ FactorInteger[#][[All, 2]] > 1 &])^2]
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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A242977
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Decimal expansion of Sum_{k>1} 1/(k*(k-1)*zeta(k)), a constant related to Niven's constant.
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+10
2
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7, 6, 6, 9, 4, 4, 4, 9, 0, 5, 2, 1, 0, 8, 8, 2, 4, 1, 6, 5, 2, 4, 1, 7, 9, 2, 3, 0, 0, 3, 1, 7, 6, 9, 3, 0, 9, 7, 4, 7, 5, 7, 8, 8, 9, 9, 3, 1, 9, 0, 5, 1, 6, 9, 6, 5, 4, 1, 2, 2, 0, 8, 1, 6, 0, 7, 8, 9, 6, 8, 4, 2, 3, 7, 5, 6, 7, 9, 5, 7, 7, 5, 7, 8, 9, 3, 7, 4, 6, 2, 9, 8, 4, 0, 9, 9, 4, 3
(list;
constant;
graph;
refs;
listen;
history;
text;
internal format)
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OFFSET
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0,1
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COMMENTS
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The asymptotic mean of the reciprocals of the maximal exponent in prime factorization of the positive integers. - Amiram Eldar, Dec 15 2022
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REFERENCES
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Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 2.6 Niven's constant, p. 113.
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LINKS
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Wolfgang Schwarz and Jürgen Spilker, A remark on some special arithmetical functions, in: E. Laurincikas , E. Manstavicius and V. Stakenas (eds.), Analytic and Probabilistic Methods in Number Theory, Proceedings of the Second International Conference in Honour of J. Kubilius, Palanga, Lithuania, 23-27 September 1996, New Trends in Probability and Statistics, Vol. 4, VSP BV & TEV Ltd. (1997), pp. 221-245.
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FORMULA
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Equals 1 + Sum_{k>=2} (1/zeta(k)-1)/(k*(k-1)). - Amiram Eldar, Dec 15 2022
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EXAMPLE
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0.766944490521088241652417923...
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MATHEMATICA
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digits = 98; m0 = 100; dm = 100; Clear[f]; f[m_] := f[m] = NSum[1/(k*(k - 1)*Zeta[k]), {k, 2, m}, WorkingPrecision -> digits + 10, NSumTerms -> m] + 1/m; f[m0]; f[m = m0 + dm]; While[RealDigits[f[m], 10, digits] != RealDigits[f[m - dm], 10, digits], Print["m = ", m ]; m = m + dm]; RealDigits[f[m], 10, digits] // First
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PROG
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(PARI) sumpos(k = 2, 1/(k*(k-1)*zeta(k))) \\ Amiram Eldar, Dec 15 2022
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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A340585
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Noncube perfect powers.
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+10
2
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4, 9, 16, 25, 32, 36, 49, 81, 100, 121, 128, 144, 169, 196, 225, 243, 256, 289, 324, 361, 400, 441, 484, 529, 576, 625, 676, 784, 841, 900, 961, 1024, 1089, 1156, 1225, 1296, 1369, 1444, 1521, 1600, 1681, 1764, 1849, 1936, 2025, 2048, 2116, 2187, 2209, 2304, 2401, 2500
(list;
graph;
refs;
listen;
history;
text;
internal format)
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OFFSET
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1,1
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COMMENTS
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This was the original definition of A239870. However, the true definition of that sequence seems to be slightly different.
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LINKS
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FORMULA
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Sum_{n>=1} 1/a(n) = 1 - zeta(3) + Sum_{k>=2} mu(k)*(1-zeta(k)) = 1 - A002117 + A072102 = 0.6724074652... - Amiram Eldar, Jan 12 2021
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MAPLE
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filter:= proc(n) local g;
g:= igcd(op(ifactors(n)[2][.., 2]));
g > 1 and (g mod 3 <> 0)
end proc:
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MATHEMATICA
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Select[Range[2, 2500], (g = GCD @@ FactorInteger[#][[;; , 2]]) > 1 && !Divisible[g, 3] &] (* Amiram Eldar, Jan 12 2021 *)
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PROG
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(PARI) for(n=2, 2500, if( ispower(n) % 3, print1(n, ", ")))
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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A242972
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Decimal expansion of a constant related to Niven's constant.
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+10
0
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8, 9, 2, 8, 9, 4, 5, 7, 1, 4, 5, 1, 2, 6, 6, 0, 9, 0, 4, 5, 7, 0, 0, 9, 4, 3, 0, 0, 2, 2, 4, 2, 7, 0, 9, 3, 3, 6, 0, 5, 0, 4, 0, 8, 5, 9, 4, 4, 5, 6, 8, 4, 3, 2, 6, 4, 7, 4, 9, 5, 6, 7, 9, 0, 7, 4, 3, 7, 2, 7, 3, 4, 3, 8, 7, 2, 7, 6, 5, 6, 4, 9, 4, 9, 0, 6, 6, 9, 6, 8, 8, 7, 3, 6, 9, 4, 1, 7, 8, 3
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OFFSET
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0,1
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REFERENCES
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Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 2.6 Niven's constant, p. 113.
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LINKS
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FORMULA
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Equals Sum_(p prime) (zeta(p)-1).
Equals Sum_{k>=2} Sum_{p prime} 1/k^p. - Amiram Eldar, Aug 21 2020
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EXAMPLE
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0.89289457145126609045700943002242709336...
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MATHEMATICA
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digits = 100; k0 = 100; dk = 50; $MaxExtraPrecision = 12*digits; z[n_?NumericQ] := Zeta[Prime[n // Floor]]; Clear[s]; s[k_] := s[k] = NSum[z[n] - 1, {n, 1, k}, WorkingPrecision -> digits + 10, NSumTerms -> 10*digits]*(1 + NSum[Zeta[n] - 1, {n, k + 1, Infinity}, WorkingPrecision -> digits + 10]); s[k0] ; s[k = k0 + dk]; While[RealDigits[s[k], 10, digits] != RealDigits[s[k - dk], 10, digits], Print["k = ", k]; k = k + dk]; RealDigits[s[k], 10, digits] // First
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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