A128468 -id:A128468 - cp's OEIS frontend

A078946 Primes p such that p, p+2, p+6, p+12 and p+14 are consecutive primes.

17, 227, 1277, 1607, 3527, 3917, 4637, 4787, 27737, 38447, 39227, 44267, 71327, 97367, 99707, 113147, 122027, 122387, 124337, 165707, 183497, 187127, 191447, 197957, 198827, 275447, 290657, 312197, 317957, 347057, 349397, 416387, 418337, 421697, 427067, 443867

Offset: 1

Labos Elemer, Dec 19 2002

nonn

			227 is in the sequence since 227, 229 = 227 + 2, 233 = 227 + 6, 239 = 227 + 12 and 241 = 227 + 14 are consecutive primes.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from T. D. Noe)
Eric Weisstein's World of Mathematics, Prime Triplet.

Subsequence of A128468.
Subsequence of A078847. - R. J. Mathar, Feb 10 2013
Cf. A001223, A078866, A078867, A078947-A078971, A022006, A022007.

[p: p in PrimesInInterval(7,1000000) | forall{i: i in [2,6,12,14] | IsPrime(p+i)}]; // Vincenzo Librandi, Apr 19 2015

Rest@ Select[Prime@ Range@ 36000, AllTrue[{2, 6, 12, 14} + #, PrimeQ] &] (* Michael De Vlieger, Apr 18 2015, Version 10 *)
Select[Partition[Prime[Range[36000]],5,1],Differences[#]=={2,4,6,2}&][[All,1]] (* Harvey P. Dale, Jun 14 2022 *)

isok(p) = isprime(p) && (nextprime(p+1)==p+2) && (nextprime(p+3)== p+6) && (nextprime(p+7)==p+12) && (nextprime(p+13)==p+14); \\ Michel Marcus, Dec 10 2013

list(lim) = {my(p1 = 2, p2 = 3, p3 = 5, p4 = 7); forprime(p5 = 11, lim, if(p2 - p1 == 2 && p3 - p2 == 4 && p4 - p3 == 6 && p5 - p4 == 2, print1(p1, ", ")); p1 = p2; p2 = p3; p3 = p4; p4 = p5);} \\ Amiram Eldar, Feb 21 2025

a(n) == 17 (mod 30). - Amiram Eldar, Feb 21 2025

Edited by Dean Hickerson, Dec 20 2002

A253624 Initial members of prime sextuples (p, p+2, p+12, p+14, p+24, p+26).

Original entry on oeis.org

5, 17, 1277, 4217, 21587, 91127, 103967, 113147, 122027, 236867, 342047, 422087, 524957, 560477, 626597, 754967, 797567, 909317, 997097, 1322147, 1493717, 1698857, 1748027, 1762907, 2144477, 2158577, 2228507, 2398157, 2580647, 2615957

Offset: 1

Karl V. Keller, Jr., Jan 06 2015

nonn

This sequence is primes p for which there exist three twin prime pairs (p, p+2), (p+12, p+14) and (p+24, p+26).
Excluding 5, this is a subsequence of each of the following: A128468 (a(n)=30n+17). A039949 (Primes of the form 30n-13), A181605 (twin primes ending in 7).
Note that not in all cases (p, p+2, p+12, p+14, p+24, p+26) are consecutive primes; the first p's for which (p, p+2, p+12, p+14, p+24, p+26) are consecutive primes are 4217, 21587, 91127, 103967, 236867, 342047, 422087, 560477, 797567, 909317, 1322147, 1493717, 1748027, 1762907, 2144477, 2158577, 2228507, 2615957 (not in OEIS). - Zak Seidov, May 16 2017

			For p = 17, the numbers 17, 19, 29, 31, 41, 43 are primes.

Karl V. Keller, Jr., Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Twin Primes
Wikipedia, Twin prime

Cf. A077800 (twin primes), A128468, A039949, A181605.

select(t -> andmap(isprime, [t,t+2,t+12,t+14,t+24,t+26]),
[5, seq(30*k+17,k=0..10^5)]); # Robert Israel, Jan 07 2015

Select[Prime@ Range[2*10^5], Times @@ Boole@ PrimeQ[# + {2, 12, 14, 24, 26}] == 1 &] (* Michael De Vlieger, May 16 2017 *)
Select[Prime[Range[200000]],AllTrue[#+{2,12,14,24,26},PrimeQ]&] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Jun 15 2021 *)

from sympy import isprime
for n in range(1,10000001,2):
  if isprime(n) and isprime(n+2) and isprime(n+12) and isprime(n+14) and isprime(n+24) and isprime(n+26): print(n,end=', ')

from sympy import isprime, primerange
def aupto(limit):
  alst = []
  for p in primerange(2, limit+1):
    if all(map(isprime, [p+2, p+12, p+14, p+24, p+26])): alst.append(p)
  return alst
print(aupto(3*10**6)) # Michael S. Branicky, May 17 2021

A267984 Numbers congruent to {17, 23} mod 30.

Original entry on oeis.org

17, 23, 47, 53, 77, 83, 107, 113, 137, 143, 167, 173, 197, 203, 227, 233, 257, 263, 287, 293, 317, 323, 347, 353, 377, 383, 407, 413, 437, 443, 467, 473, 497, 503, 527, 533, 557, 563, 587, 593, 617, 623, 647, 653, 677, 683, 707, 713, 737, 743, 767, 773

Offset: 1

Arkadiusz Wesolowski, Jan 23 2016

Union of A128468 and A128473.

For all k >= 1 the numbers 2^k + a(n) and a(n)*2^k + 1 do not form a pair of primes, where n is any positive integer.

Colin Barker, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (1,1,-1).

Cf. A128468, A128473, A267985.

[n: n in [0..773] | n mod 30 in {17, 23}];

LinearRecurrence[{1, 1, -1}, {17, 23, 47}, 52]

Vec(x*(17 + 6*x + 7*x^2)/((1 + x)*(1 - x)^2) + O(x^53))

a(n) = a(n-1) + a(n-2) - a(n-3), n >= 4.
G.f.: x*(17 + 6*x + 7*x^2)/((1 + x)*(1 - x)^2).
a(n) = a(n-2) + 30.
a(n) = 10*(3*n - 2) - a(n-1).
From Colin Barker, Jan 24 2016: (Start)
a(n) = (30*n - 9*(-1)^n - 5)/2 for n>0.
a(n) = 15*n - 7 for n>0 and even.
a(n) = 15*n + 2 for n odd.
(End)
E.g.f.: 7 + ((30*x - 5)*exp(x) - 9*exp(-x))/2. - David Lovler, Sep 10 2022

A375645 Products of prime 7-tuples (p, p+2, p+8, p+12, p+14, p+18, p+20) where p = A022010(n).

Original entry on oeis.org

183698727318433150098859517, 43573095131179423946916455382173477, 151752127452301913425377267345374694407, 37933916719513692044984369353553394500687, 336012768546310957228958479424678156040797, 2608471791567290523882206574758483434858457, 523352977400310485591027030692542102863968347677

Offset: 1

Michael De Vlieger, Aug 23 2024

nonn

Primes p in A022010 belong to 179 (mod 210), therefore a(n) is congruent to the product of residues {179, 181, 187, 191, 193, 197, 199} (mod 210), so a(n) is congruent to 17 (mod 210). Gaps between prime factors are {2, 6, 4, 2, 4, 2}.

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A022010, A128468, A375645.

Map[Times @@ NextPrime[#, Range[0, 6]] &, Select[Prime@ Range[2^20], AllTrue[# + {2, 8, 12, 14, 18, 20}, PrimeQ] &]]

A245568 Initial members of prime quadruples (n, n+2, n+24, n+26).

Original entry on oeis.org

5, 17, 617, 857, 1277, 1427, 1697, 2087, 2687, 3557, 4217, 5417, 5477, 7307, 8837, 9437, 10067, 13877, 17657, 18287, 20747, 21587, 23537, 25577, 27917, 28547, 30467, 32117, 32297, 35507, 37337, 37547, 40127, 41177, 41387, 41957

Offset: 1

Karl V. Keller, Jr., Jan 09 2015

nonn

This sequence is prime n, where there exist two twin prime pairs of (n, n+2, n+24, n+26).
Excluding 5, this is a subsequence of each of the following: A128468 (a(n) = 30*n + 17), A039949 (Primes of the form 30n-13), A181605 (twin primes ending in 7).
A253624 is a subsequence of this sequence.

			For n = 17, the numbers 17, 19, 41, 43 are primes.

Karl V. Keller, Jr., Table of n, a(n) for n = 1..100000
Eric Weisstein's World of Mathematics, Prime Quadruplet.
Eric Weisstein's World of Mathematics, Twin Primes
Wikipedia, Twin prime

Cf. A077800 (twin primes), A128468, A039949, A181605, A253624.

a245568[n_] := Select[Prime@ Range@ n, And[PrimeQ[# + 2], PrimeQ[# + 24], PrimeQ[# + 26]] &]; a245568[5000] (* Michael De Vlieger, Jan 11 2015 *)

from sympy import isprime
for n in range(1,10000001,2):
  if isprime(n) and isprime(n+2) and isprime(n+24) and isprime(n+26): print(n,end=', ')

A248474 Numbers congruent to 13 or 17 mod 30.

Original entry on oeis.org

13, 17, 43, 47, 73, 77, 103, 107, 133, 137, 163, 167, 193, 197, 223, 227, 253, 257, 283, 287, 313, 317, 343, 347, 373, 377, 403, 407, 433, 437, 463, 467, 493, 497, 523, 527, 553, 557, 583, 587, 613, 617, 643, 647, 673, 677, 703, 707, 733, 737, 763, 767, 793, 797

Offset: 1

Karl V. Keller, Jr., Oct 06 2014

The combination of A082369(30*n+13) and A128468(30*n+17) is the base sequence for A140533(Primes congruent to 13 or 17 mod 30).

Karl V. Keller, Jr., Table of n, a(n) for n = 1..10000
Index entries for linear recurrences with constant coefficients, signature (1,1,-1).

Cf. A082369 (30*n+13), A128468 (30*n+17).

Cf. A039949 (Primes of the form 30n-13), A132233 (Primes congruent to 13 mod 30), A140533 (Primes congruent to 13 or 17 mod 30).

Flatten[Table[{15n - 2, 15n + 2}, {n, 1, 41, 2}]] (* Alonso del Arte, Oct 06 2014 *)

Vec(x*(13*x^2+4*x+13)/((x-1)^2*(x+1)) + O(x^100)) \\ Colin Barker, Oct 07 2014

for n in range(1,101):
  print (n*30-17),
  print (n*30-13),

From Colin Barker, Oct 07 2014: (Start)
a(n) = (-15-11*(-1)^n+30*n)/2.
a(n) = a(n-1)+a(n-2)-a(n-3).
G.f.: x*(13*x^2+4*x+13) / ((x-1)^2*(x+1)). (End)
E.g.f.: 13 + ((30*x - 15)*exp(x) - 11*exp(-x))/2. - David Lovler, Sep 10 2022
Sum_{n>=1} (-1)^(n+1)/a(n) = (sqrt(2*(5+sqrt(5)))+sqrt(3)-sqrt(15))*Pi / (30*(sqrt(6*(5+sqrt(5)))+sqrt(5)-1)). - Amiram Eldar, Jul 30 2024

A248523 Initial members of prime quadruples (n, n+2, n+144, n+146).

Original entry on oeis.org

5, 137, 1787, 1997, 2237, 2657, 3527, 4127, 4337, 4787, 8087, 12107, 13757, 14447, 17987, 19697, 21377, 23057, 23687, 31247, 32297, 34157, 34367, 35447, 37547, 38567, 39227, 43397, 48677, 51197, 51827, 53087, 58907, 65027, 65837

Offset: 1

Karl V. Keller, Jr., Jan 11 2015

nonn

This sequence is prime n, where there exist two twin prime pairs of (n,n+2), (n+144,n+146).

Excluding 5, this is a subsequence of each of the following: A128468 (a(n)=30*n+17), A039949 (Primes of the form 30n-13), A181605 (twin primes ending in 7).

			For n=137, the numbers 137, 139, 281, 283, are primes.

Karl V. Keller, Jr., Table of n, a(n) for n = 1..100000
Eric Weisstein's World of Mathematics, Prime Quadruplet.
Eric Weisstein's World of Mathematics, Twin Primes
Wikipedia, Twin prime

Cf. A077800 (twin primes), A128468, A039949, A181605.

from sympy import isprime
for n in range(1,10000001,2):
    if isprime(n) and isprime(n+2) and isprime(n+144) and isprime(n+146): print(n,end=', ')

A248661 Initial members of prime quadruples (n, n+2, n+54, n+56).

Original entry on oeis.org

5, 17, 137, 227, 827, 1427, 1667, 1877, 2027, 2087, 2657, 3527, 3767, 4217, 4967, 10037, 11117, 11777, 12107, 13877, 17987, 19697, 20717, 21557, 22037, 23687, 24977, 27527, 27737, 34157, 37307, 41177, 42017, 42407, 47657, 48677

Offset: 1

Karl V. Keller, Jr., Jan 11 2015

nonn

This sequence is prime n, where there exist two twin prime pairs of (n,n+2), (n+54,n+56).

Excluding 5, this is a subsequence of each of the following: A128468 (a(n)=30*n+17), A039949 (primes, 30n-13), A181605 (twin primes, end 7), and A092340 (prime n, where n^2+2*n divides (fibonacci(n^2)+fibonacci(2*n))).

			For n=17, the numbers 17, 19, 71, 73, are primes.

Karl V. Keller, Jr., Table of n, a(n) for n = 1..100000
Eric Weisstein's World of Mathematics, Prime Quadruplet.
Eric Weisstein's World of Mathematics, Twin Primes
Wikipedia, Twin prime

Cf. A077800 (twin primes), A128468, A039949, A181605, A092340.

from sympy import isprime
for n in range(1,10000001,2):
  if isprime(n) and isprime(n+2) and isprime(n+54) and isprime(n+56): print(n,end=', ')

cp's OEIS Frontend

A078946 Primes p such that p, p+2, p+6, p+12 and p+14 are consecutive primes.

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A253624 Initial members of prime sextuples (p, p+2, p+12, p+14, p+24, p+26).

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A267984 Numbers congruent to {17, 23} mod 30.

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A375645 Products of prime 7-tuples (p, p+2, p+8, p+12, p+14, p+18, p+20) where p = A022010(n).

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A245568 Initial members of prime quadruples (n, n+2, n+24, n+26).

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A248474 Numbers congruent to 13 or 17 mod 30.

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A248523 Initial members of prime quadruples (n, n+2, n+144, n+146).

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