Search: a081357 -id:a081357
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60, 84, 90, 96, 108, 126, 132, 140, 150, 156, 160, 180, 198, 204, 220, 224, 228, 234, 240, 252, 260, 276, 294, 300, 306, 308, 315, 336, 340, 342, 348, 350, 352, 360, 364, 372, 380, 396, 414, 416, 420, 432, 444, 460, 476, 480, 486, 490, 492, 495, 500, 504, 516
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OFFSET
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1,1
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COMMENTS
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Analogous to sublime numbers (A081357), with abundant numbers instead of perfect numbers.
The least odd term is a(27) = 315 and the least term that is coprime to 6 is a(298) = 1925.
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REFERENCES
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József Sándor and E. Egri, Arithmetical functions in algebra, geometry and analysis, Advanced Studies in Contemporary Mathematics, Vol. 14, No. 2 (2007), pp. 163-213.
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LINKS
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EXAMPLE
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60 is a term since both d(60) = 12 and sigma(60) = 168 are abundant numbers: sigma(12) = 28 > 2*12 = 24 and sigma(168) = 480 > 2*168 = 336.
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MATHEMATICA
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abQ[n_] := DivisorSigma[1, n] > 2*n; nobAbQ[n_] := And @@ abQ /@ DivisorSigma[{0, 1}, n]; Select[Range[500], nobAbQ]
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PROG
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(PARI) isab(k) = sigma(k) > 2*k; \\ A005101
isok(k) = my(f=factor(k)); isab(numdiv(f)) && isab(sigma(f)); \\ Michel Marcus, Dec 02 2021
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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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A349759
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Nobly deficient numbers: numbers k such that both d(k) = A000005(k) and sigma(k) = A000203(k) are deficient numbers (A005100).
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+10
6
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1, 2, 3, 4, 7, 8, 9, 13, 16, 21, 25, 31, 36, 37, 43, 48, 49, 61, 64, 67, 73, 81, 93, 97, 100, 109, 111, 112, 121, 127, 128, 144, 151, 157, 162, 163, 169, 181, 183, 192, 193, 196, 208, 211, 217, 219, 225, 229, 241, 256, 277, 283, 289, 313, 324, 331, 337, 361, 373
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OFFSET
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1,2
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COMMENTS
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Analogous to sublime numbers (A081357), with deficient numbers instead of perfect numbers.
If p != 5 is a prime such that (p+1)/2 is also a prime (i.e., p is in A005383 \ {5}), then p is a term of this sequence.
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LINKS
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EXAMPLE
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2 is a term since both d(2) = 2 and sigma(2) = 3 are deficient numbers.
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MATHEMATICA
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defQ[n_] := DivisorSigma[1, n] < 2*n; nobDefQ[n_] := And @@ defQ /@ DivisorSigma[{0, 1}, n]; Select[Range[400], nobDefQ]
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PROG
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(PARI) isdef(k) = sigma(k) < 2*k; \\ A005100
isok(k) = my(f=factor(k)); isdef(numdiv(f)) && isdef(sigma(f)); \\ Michel Marcus, Dec 03 2021
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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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A146542
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Numbers m such that sigma(m) is a perfect number.
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+10
2
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5, 12, 427, 10924032, 16125952, 22017387, 24376323, 32501857, 33288097, 3757637632, 6241076643, 8522760577, 45091651584, 66563866624, 86692869921, 137421905953, 137437511683, 727145809044307968, 1152771972099211264, 845044701535107443245558061611352064
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OFFSET
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1,1
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LINKS
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EXAMPLE
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The divisors of 5 are 1 and 5, which add up to 6. 6 is a perfect number because its proper divisors are 1, 2 and 3, which also add up to 6.
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MAPLE
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with(numtheory); P:=proc(q) local n; for n from 1 to q do
if sigma(sigma(n))=2*sigma(n) then print(n);
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PROG
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(PARI) isok(n) = sigma(sigma(n)) == 2*sigma(n); \\ Michel Marcus, Oct 22 2013
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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Howard Berman (howard_berman(AT)hotmail.com), Oct 31 2008
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EXTENSIONS
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STATUS
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approved
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A233482
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Numbers for which the number of divisors and the sum of the distinct prime divisors are both perfect.
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+10
2
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575, 2057, 2645, 3179, 4416, 8512, 12275, 33534, 94272, 138431, 203075, 218176, 392747, 715878, 918592, 982157, 991841, 1082176, 1205405, 1244387, 1559616, 1690432, 1966912, 2344079, 2464576, 2982976, 3386176, 3452992, 3625792, 3821632, 3867712, 3900497
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graph;
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listen;
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OFFSET
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1,1
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COMMENTS
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LINKS
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EXAMPLE
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575 is in the sequence because tau(575) = 6 and sopf(575) = 28,
4416 is in the sequence because tau(4416) = 28 and sopf(4416) = 28,
12275 is in the sequence because tau(12275) = 6 and sopf(12275) = 496,
203075 is in the sequence because tau(203075) = 6 and sopf(203075) = 8128.
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MAPLE
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with(numtheory): lst:={6, 28, 496, 8128, 33550336, 8589869056, 137438691328, 2305843008139952128, 2658455991569831744654692615953842176, 191561942608236107294793378084303638130997321548169216} :n1:=nops(lst): for n from 1 to 1000000 do :x:=factorset(n):n2:=nops(x): s:=sum('x[i]', 'i'=1..n2):
ii:=0:for m from 1 to n1 do:if s=lst[m] then ii:=1:else fi:od:jj:=0:for p from 1 to n1 do:if tau(n)=lst[p] then jj:=1:else fi:od:if ii=1 and jj=1 then printf(`%d, `, n):else fi:od:
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MATHEMATICA
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Select[Range[4*10^6], AllTrue[{DivisorSigma[0, #], Total[FactorInteger[#][[All, 1]]]}, PerfectNumberQ]&] (* Harvey P. Dale, Aug 11 2021 *)
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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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A233563
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Numbers for which the number of prime divisors counted with multiplicity and the sum of the distinct prime divisors are both perfect.
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+10
1
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1104, 1656, 2128, 2484, 3726, 4620, 6930, 7448, 11550, 12285, 12696, 16170, 19044, 20216, 20475, 23568, 25410, 26068, 28566, 28665, 34125, 35352, 47775, 53028, 53235, 54544, 66885, 70756, 71875, 79542, 88725, 91238, 124215, 146004, 190904, 192052, 201180
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OFFSET
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1,1
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COMMENTS
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LINKS
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EXAMPLE
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1104 is in the sequence because bigomega(1104) = 6 and sopf(1104) = 28,
23568 is in the sequence because bigomega(23568) = 6 and sopf(23568) = 496,
389904 is in the sequence because bigomega(389904) = 6 and sopf(389904) = 8128.
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MAPLE
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with(numtheory): lst:={6, 28, 496, 8128, 33550336, 8589869056, 137438691328, 2305843008139952128, 2658455991569831744654692615953842176, 191561942608236107294793378084303638130997321548169216} :n1:=nops(lst): for n from 1 to 1000000 do :x:=factorset(n):n2:=nops(x): s:=sum('x[i]', 'i'=1..n2):
ii:=0:for m from 1 to n1 do:if s=lst[m] then ii:=1:else fi:od:jj:=0:for p from 1 to n1 do:if bigomega(n)=lst[p] then jj:=1:else fi:od:if ii=1 and jj=1 then printf(`%d, `, n):else fi:od:
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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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A290149
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Totient sublime numbers: numbers k such that the number of terms in the iterations of phi(k) from k to 1, A032358(k)+2, and their sum, A092693(k) are both perfect totient numbers (A082897).
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+10
0
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OFFSET
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1,1
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COMMENTS
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Analogous to A081357 (sublime numbers), as A082897 (perfect totient numbers) is analogous to A000396 (perfect numbers).
No other terms below 10^8.
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LINKS
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EXAMPLE
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There are 9 terms in the iterations of phi(k) for 2916: 2916, 972, 324, 108, 36, 12, 4, 2, 1. Their sum is 4375. Both 9 and 4375 are perfect totient numbers (A082897).
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MATHEMATICA
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iterList [n_] := FixedPointList[EulerPhi@# &, n]; sumIter [n_] := Plus @@ iterList[n] - 1; numIter[n_] := Length[iterList[n]] - 1; perfTotQ[n_] := sumIter[n] == 2 n; totSublimeQ[n_] := perfTotQ[numIter[n]] && perfTotQ[sumIter[n]]; Select[Range [10^8], totSublimeQ]
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CROSSREFS
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KEYWORD
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nonn,more
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AUTHOR
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STATUS
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approved
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