Displaying 1-10 of 45 results found.
4, 5, 7, 8, 10, 20, 22, 35, 50, 52, 53, 55, 97, 98, 113, 115, 140, 155, 157, 175, 230, 232, 308, 322, 412, 413, 428, 430, 440, 442, 545, 547, 640, 650, 652, 713, 715, 725, 742, 743, 745, 805, 833, 848, 893, 935, 937, 938, 998, 1000, 1042, 1043, 1070, 1120, 1135
-1, 1, -1, 1, -1, 1, -1, 1, 1, -1, 1, -1, -1, 1, 1, -1, 1, 1, -1, -1, 1, -1, 1, -1, -1, 1, 1, -1, 1, -1, 1, -1, -1, 1, -1, 1, -1, 1, -1, 1, -1, -1, 1, 1, 1, 1, -1, 1, 1, -1, -1, 1, 1, -1, -1, 1, -1, 1, -1, 1, 1, 1, -1, 1, 1, -1, 1, -1, -1, 1, -1, -1, -1, 1, 1, -1, -1, 1, -1, -1, -1, 1, -1, 1, 1, -1
Primes p such that p + 6 is also prime. (Lesser of a pair of sexy primes.)
+10
117
5, 7, 11, 13, 17, 23, 31, 37, 41, 47, 53, 61, 67, 73, 83, 97, 101, 103, 107, 131, 151, 157, 167, 173, 191, 193, 223, 227, 233, 251, 257, 263, 271, 277, 307, 311, 331, 347, 353, 367, 373, 383, 433, 443, 457, 461, 503, 541, 557, 563, 571, 587, 593, 601, 607, 613, 641, 647
LINKS
Eric Weisstein's World of Mathematics, Sexy Primes. [The definition in this webpage is unsatisfactory, because it defines a "sexy prime" as a pair of primes.- N. J. A. Sloane, Mar 07 2021].
MAPLE
option remember;
if n = 1 then
5;
else
for a from procname(n-1)+2 by 2 do
if isprime(a) and isprime(a+6) then
return a;
end if;
end do:
end if;
MATHEMATICA
Select[Prime[Range[120]], PrimeQ[#+6]&] (* Harvey P. Dale, Mar 20 2018 *)
PROG
(Magma) [n: n in [0..40000] | IsPrime(n) and IsPrime(n+6)] // Vincenzo Librandi, Aug 04 2010 (Haskell)
a023201 n = a023201_list !! (n-1)
a023201_list = filter ((== 1) . a010051 . (+ 6)) a000040_list
Initial members of prime triples (p, p+2, p+6).
+10
81
5, 11, 17, 41, 101, 107, 191, 227, 311, 347, 461, 641, 821, 857, 881, 1091, 1277, 1301, 1427, 1481, 1487, 1607, 1871, 1997, 2081, 2237, 2267, 2657, 2687, 3251, 3461, 3527, 3671, 3917, 4001, 4127, 4517, 4637, 4787, 4931, 4967, 5231, 5477
MAPLE
if n= 1 then
5;
else
for a from procname(n-1)+2 by 2 do
if isprime(a) and isprime(a+2) and isprime(a+6) then
return a;
end if;
end do:
end if;
MATHEMATICA
Transpose[Select[Partition[Prime[Range[1000]], 3, 1], Differences[#]=={2, 4}&]][[1]] (* Harvey P. Dale, Dec 24 2011 *)
PROG
(Magma) [ p: p in PrimesUpTo(10000) | IsPrime(p+2) and IsPrime(p+6) ] // Vincenzo Librandi, Nov 19 2010 (Python)
from sympy import primerange
def aupto(limit):
p, q, alst = 2, 3, []
for r in primerange(5, limit+7):
if p+2 == q and p+6 == r: alst.append(p)
p, q = q, r
return alst
Initial members of prime triples (p, p+4, p+6).
+10
61
7, 13, 37, 67, 97, 103, 193, 223, 277, 307, 457, 613, 823, 853, 877, 1087, 1297, 1423, 1447, 1483, 1663, 1693, 1783, 1867, 1873, 1993, 2083, 2137, 2377, 2683, 2707, 2797, 3163, 3253, 3457, 3463, 3847, 4153, 4513, 4783, 5227, 5413, 5437, 5647, 5653, 5737, 6547
PROG
(Magma) [p: p in PrimesUpTo(10000) | IsPrime(p+4) and IsPrime(p+6)]; // Vincenzo Librandi, Aug 23 2015
Primes followed by a gap of 6, i.e., next prime is p + 6.
+10
58
23, 31, 47, 53, 61, 73, 83, 131, 151, 157, 167, 173, 233, 251, 257, 263, 271, 331, 353, 367, 373, 383, 433, 443, 503, 541, 557, 563, 571, 587, 593, 601, 607, 647, 653, 677, 727, 733, 751, 941, 947, 971, 977, 991, 1013, 1033, 1063, 1097, 1103, 1117, 1123, 1181
COMMENTS
Original name: Lower prime of a difference of 6 between consecutive primes.
Conjecture: The sequence is infinite and for every n >= 7746, a(n+1) < a(n)^(1+1/n). Namely for n >= 7746, a(n)^(1/n) is a strictly decreasing function of n (See comment lines of the sequence A248855). - Jahangeer Kholdi and Farideh Firoozbakht, Nov 29 2014
EXAMPLE
23 is a term as the next prime 29 = 23 + 6.
MAPLE
option remember;
if n = 1 then
return 23;
else
p := nextprime(procname(n-1)) ;
q := nextprime(p) ;
while q-p <> 6 do
p := q ;
q := nextprime(p) ;
end do:
return p;
end if;
MATHEMATICA
Transpose[Select[Partition[Prime[Range[200]], 2, 1], Last[#] - First[#] == 6 &]][[1]] (* Bruno Berselli, Apr 09 2013 *)
PROG
(PARI) apply( A031924(n, p=2, show=0, g=6)={forprime(q=p+1, , p+g!=(p=q) || (show&&print1(p-g", ")) || n-- || return(p-g))}, [1..99]) \\ Use nxt(p)= A031924(1, p) to get the term following p, use show=1 to print all a(1..n), g to select a different gap. - M. F. Hasler, Jan 02 2020 (Magma) [p: p in PrimesUpTo(1200) | NextPrime(p)-p eq 6]; // Bruno Berselli, Apr 09 2013 (GAP) P:=Filtered([1..1200], IsPrime);;
List(Filtered([1..Length(P)-1], i->P[i+1]-P[i]=6), k->P[k]); # Muniru A Asiru, Jan 30 2019
Primes p such that three (the maximum number) primes occur between p and p+12.
+10
31
5, 7, 11, 97, 101, 1481, 1867, 3457, 5647, 15727, 16057, 16061, 19417, 19421, 21011, 22271, 43777, 43781, 55331, 79687, 88807, 101107, 144161, 165701, 166841, 195731, 201821, 225341, 247601, 257857, 266677, 268811, 276037, 284737, 326141, 340927
COMMENTS
A086140 is the union of A022006 and A022007. By merging the two b-files I have extended the current b-file up to n=10000 (nearly n=20000 would have been possible). I add a comparison (see Links) between the frequency of prime 5-tuples and an asymptotic approximation, which is unproven but likely to be true, and based on a conjecture first published by Hardy and Littlewood in 1923. Twins, triples and quadruplets are treated as well. - Gerhard Kirchner, Dec 07 2016
EXAMPLE
There are two types of prime 5-tuples, and both are represented in this sequence. (11, 13, 17, 19, 23) is a prime 5-tuple of the form (p, p+2, p+6, p+8, p+12), so 11 is in the sequence, and (97, 101, 103, 107, 109) is a prime 5-tuple of the form (p, p+4, p+6, p+10, p+12), so 97 is in the sequence. - Michael B. Porter, Dec 19 2016
MATHEMATICA
cp[x_, y_] := Count[Table[PrimeQ[i], {i, x, y}], True] {d=12, k=0}; Do[s=Prime[n]; s1=Prime[n+1]; If[PrimeQ[s+d]&&Equal[cp[s+1, s+d-1], 3], k=k+1; Print[s]], {n, 1, 100000}]
(* Second program: *)
Transpose[Select[Partition[Prime[Range[30000]], 5, 1], #[[5]]-#[[1]] == 12&]][[1]] (* Harvey P. Dale, Jun 11 2015 *)
Initial members of prime septuplets.
+10
29
11, 5639, 88799, 165701, 284729, 626609, 855719, 1068701, 1146779, 6560999, 7540439, 8573429, 11900501, 15760091, 17843459, 18504371, 19089599, 21036131, 24001709, 25658441, 39431921, 42981929, 43534019, 45002591, 67816361, 69156539, 74266259, 79208399, 80427029, 84104549, 86818211, 87988709, 93625991, 124066079
CROSSREFS
Initial members of all of the first prime k-tuplets:
Cf. A343637 (distance from 10^n to the next septuplet).
Initial members of prime 9-tuplets (or nonuplets).
+10
28
7, 11, 13, 17, 1277, 88789, 113143, 113147, 855709, 74266249, 182403491, 226449521, 252277007, 408936947, 521481197, 626927443, 910935911, 964669609, 1042090781, 1116452627, 1209950867, 1422475909, 1459270271, 1645175087, 2117861719, 2335215973, 2558211559, 2843348351, 2873599429, 2966003057, 3447123283, 3947480417
COMMENTS
Primes prime(m) such that prime(m+8) = prime(m) + 30. - Zak Seidov, Jul 06 2015
MATHEMATICA
{p, q, r, s, t, u, v, w, x} = Prime@ Range@ 9; lst = {}; While[p < 1000000001, If[p + 30 == x, AppendTo[lst, p]; Print@ p]; {p, q, r, s, t, u, v, w, x} = {q, r, s, t, u, v, w, x, NextPrime@ x}]; lst (* Robert G. Wilson v, Jul 06 2015 *) Select[Partition[Prime[Range[5 10^6]], 9, 1], #[[1]]+30==#[[9]]&][[;; , 1]] (* The program generates the first 10 terms of the sequence. To generate more, increase the Range constant. *) (* Harvey P. Dale, Jul 01 2024 *)
PROG
(PARI) main(size)=v=vector(size); i=0; m=1; while(i<size, if(prime(m)+30===prime(m+8), v[i++]=prime(m)); m++; ); v \\ Anders Hellström, Jul 08 2015 (Magma) [NthPrime(n): n in [0..2*10^4] | NthPrime(n+8) eq (NthPrime(n) + 30)]; // Vincenzo Librandi, Jul 08 2015
CROSSREFS
Initial members of all of the first prime k-tuplets:
Initial members of prime 10-tuplets (or decaplets).
+10
28
11, 9853497737, 21956291867, 22741837817, 33081664151, 83122625471, 164444511587, 179590045487, 217999764107, 231255798857, 242360943257, 294920291201, 573459229151, 663903555851, 666413245007, 688697679401, 696391309697, 730121110331, 867132039857, 974275568237, 976136848847, 1002263588297
CROSSREFS
Initial members of all of the first prime k-tuplets:
Search completed in 0.050 seconds
|