A172454 -id:A172454 - cp's OEIS frontend

A191456 Primes p such that the polynomial x^2+x+p generates only primes for x=1..9.

11, 17, 41, 844427, 51448361, 86966771, 122983031, 180078317, 960959381, 1278189947, 1761702947, 1829187287, 2426256797, 2911675511, 3013107257, 4778888351, 5221343711

Offset: 1

Zak Seidov, Jun 02 2011

nonn

Charles R Greathouse IV, Table of n, a(n) for n = 1..2713

Generates primes for x=1..k: A001359 (1), A022004 (2), A172454 (3), A187057 (4), A187058 (5), A144051 (6), A187060 (7), A190800 (8), this sequence (9), A191457 (10), A191458 (11), A253592 (12), A253605 (13). Each is by definition a subsequence of preceding sequences.
Subsequence such that x=10 gives a composite number: A211238.
Cf. A160548, A164926.

is(n)=for(x=0,9, if(!isprime(x^2+x+n), return(0))); 1 \\ Charles R Greathouse IV, Sep 14 2015

A172456 Primes p such that (p, p+2, p+6, p+12, p+14, p+20) is a prime sextuple.

Original entry on oeis.org

17, 1277, 1607, 3527, 4637, 71327, 97367, 113147, 191447, 290657, 312197, 416387, 418337, 421697, 450797, 566537, 795647, 886967, 922067, 1090877, 1179317, 1300127, 1464257, 1632467, 1749257, 1866857, 1901357, 2073347, 2322107

Offset: 1

Michel Lagneau, Feb 03 2010

nonn

The last digit of each of these prime numbers is 7.
Subsequence of A078946.
The primes are always consecutive: The few ways of inserting other primes are: (p,p+2,p+4)... [impossible since one of these would be a multiple of 3]; (p,p+2,p+6),(p+8),(p+12),(p+14) [impossible since one of these would be a multiple of 5]; (p,p+2,p+6),(p+10) [impossible since one of these would be a multiple of 3]; (p,p+2,p+6),(p+12),(p+14),(p+16) [impossible since one of these would be a multiple of 3]; (p,p+2,p+6),(p+12),(p+14),(p+18) [impossible since one of these would be a multiple of 5]. - R. J. Mathar, Jun 15 2013

			The first two terms correspond to the sextuples (17,19,23,29,31,37) and (1277,1279,1283,1289,1291,1297).

R. K. Guy, Unsolved Problems in Number Theory, E30.

Amiram Eldar, Table of n, a(n) for n = 1..1000
G. E. Andrews, MacMahon's prime numbers of measurement, Amer. Math. Monthly, 82 (1975), 922-923.
T. Forbes, Prime k-tuplets
R. L. Graham and C. B. A. Peck, Problem E1910, Amer. Math. Monthly, 75 (1968), 80-81.
P. A. MacMahon, The prime numbers of measurement on a scale, Proc. Camb. Phil. Soc. 21 (1923), 651-654; reprinted in Coll. Papers I, pp. 797-800.
Eric Weisstein's World of Mathematics, Prime Triplet.

Initial members of prime quadruples (p, p+2, p+6, p+12): A172454.

Cf. A073648, A098412.

for n from 1 by 2 to 400000 do; if isprime(n) and isprime(n+2) and isprime(n+6) and isprime(n+12) and isprime(n + 14) and isprime(n+20) then print(n) else fi;od;

Select[Prime[Range[171000]],And@@PrimeQ[{#+2,#+6,#+12,#+14,#+20}]&] (* Harvey P. Dale, Jul 23 2011 *)
Select[Prime[Range[171000]],AllTrue[#+{2,6,12,14,20},PrimeQ]&] (* or *) Select[ Partition[Prime[Range[171000]],6,1],Differences[#]=={2,4,6,2,6}&][[All,1]] (* Harvey P. Dale, Sep 04 2022 *)

A291635 Number of ways to write n as x^2 + y^2 + z^2 + w^2 with x,y,z,w nonnegative integers such that p = x + 2y + 5z, p - 2, p + 4 and p + 10 are all prime.

Original entry on oeis.org

0, 1, 1, 0, 1, 3, 1, 0, 1, 2, 2, 0, 3, 7, 3, 0, 4, 4, 1, 0, 4, 7, 3, 0, 3, 5, 2, 0, 4, 6, 2, 0, 2, 3, 3, 0, 4, 8, 3, 0, 5, 8, 2, 0, 2, 5, 2, 0, 5, 8, 4, 0, 4, 5, 2, 0, 5, 6, 4, 0, 1, 8, 5, 0, 3, 9, 3, 0, 6, 8, 3, 0, 5, 13, 5, 0, 9, 9, 2, 0, 4, 6, 6, 0, 7, 11, 4, 0, 8, 10, 5, 0, 2, 11, 5, 0, 3, 10, 4, 0

Offset: 1

Zhi-Wei Sun, Aug 28 2017

Conjecture: a(n) > 0 for all n > 1 not divisible by 4.
This is stronger than the conjecture in A291624. Obviously, it implies that there are infinitely many prime quadruples (p-2, p, p+4, p+10).
We have verified that a(n) > 0 for any integer 1 < n < 10^7 not divisible by 4.

			a(61) = 1 since 61 = 4^2 + 0^2 + 3^2 + 6^2 with 4 + 2*0 + 5*3 = 19, 19 - 2 = 17, 19 + 4 = 23 and 19 + 10 = 29 all prime.
a(253) = 1 since 253 = 12^2 + 8^2 + 3^2 + 6^2 with 12 + 2*8 + 5*3 = 43, 43 - 2 = 41, 43 + 4 = 47 and 43 + 10 = 53 all prime.
a(725) = 1 since 725 = 7^2 + 0^2 + 0^2 + 26^2 with 7 + 2*0 + 5*0 = 7, 7 - 2 = 5, 7 + 4 = 11 and 7 + 10 = 17 all prime.
a(1511) = 1 since 1511 = 18^2 + 15^2 + 11^2 + 29^2 with 18 + 2*15 + 5*11 = 103, 103 - 2 = 101, 103 + 4 = 107 and 103 + 10 = 113 all prime.

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
Zhi-Wei Sun, Refining Lagrange's four-square theorem, J. Number Theory 175(2017), 167-190.
Zhi-Wei Sun, Restricted sums of four squares, arXiv:1701.05868 [math.NT], 2017.

Cf. A000040, A000118, A000290, A022004, A172454, A271518, A281976, A290935, A291150, A291191, A291455, A291624.

SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]];
PQ[p_]:=PQ[p]=PrimeQ[p]&&PrimeQ[p-2]&&PrimeQ[p+4]&&PrimeQ[p+10]
Do[r=0;Do[If[SQ[n-x^2-y^2-z^2]&&PQ[x+2y+5z],r=r+1],{x,0,Sqrt[n]},{y,0,Sqrt[n-x^2]},{z,0,Sqrt[n-x^2-y^2]}];Print[n," ",r],{n,1,100}]

A341267 Gaps between first elements of quadruple primes of the form {p, p+2, p+6, p+12}.

Original entry on oeis.org

6, 6, 24, 60, 126, 120, 294, 450, 186, 150, 54, 6, 120, 1080, 840, 390, 84, 126, 510, 150, 144, 3300, 1230, 870, 1446, 330, 1794, 726, 1434, 3360, 1326, 264, 546, 714, 1470, 1836, 1104, 30, 1026, 204, 336, 744, 2226, 810, 240, 1050, 270, 1914, 60, 876, 1980

Offset: 1

James S. DeArmon, Feb 07 2021

nonn

Primes in the quadruple need not be sequential primes.

			The first 6 quadruples are (5,7,11,17), (11,13,17,23), (17,19,23,29), (41,43,47,53), (101,103,107,113), (227,229,233,239), so the first 5 terms of the sequence are 11-5=6, 17-11=6, 41-17=24, 101-41=60, 227-101=126.

Harvey P. Dale, Table of n, a(n) for n = 1..1000
James S. DeArmon, Perl program
Eric Weisstein's World of Mathematics, Prime Triplet

Cf. A172454.

b:= proc(n) option remember; local p; p:= `if`(n=1, 1, b(n-1));
      do p:= nextprime(p);
         if andmap(isprime, [p+2, p+6, p+12]) then return p fi
      od
    end:
a:= n-> b(n+1)-b(n):
seq(a(n), n=1..65);  # Alois P. Heinz, Feb 14 2021

Differences[Select[Prime[Range[5000]],AllTrue[#+{2,6,12},PrimeQ]&]] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, May 21 2021 *)

Perl
```
# See DeArmon link.
```

a(n) = A172454(n+1) - A172454(n).

A385035 Primes p such that p + 8, p + 14, p + 18 and p + 20 are also primes.

Original entry on oeis.org

23, 53, 89, 263, 599, 1283, 1979, 3449, 5399, 5639, 11813, 14543, 41213, 42443, 44249, 47129, 55799, 57773, 65699, 74699, 75983, 79613, 84299, 87539, 88643, 88793, 88799, 113153, 115763, 126473, 143813, 148913, 150203, 160073, 163973, 167099, 176489, 178799, 178889, 209249

Offset: 1

Alexander Yutkin, Jun 15 2025

nonn

			p=23: 23+8=31, 23+14=37, 23+18=41, 23+20=43 —> prime quintuple: (23, 31, 37, 41, 43).

Cf. A000040.

Cf. A172454 [2, 4, 6], A078855 [6, 4, 2], A187057 [2, 4, 6, 8].

[p: p in PrimesUpTo(300000) | IsPrime(p+8) and IsPrime(p+14) and IsPrime(p+18) and IsPrime(p+20)]; // Vincenzo Librandi, Jul 04 2025

q:= p-> andmap(i-> isprime(p+i), [0, 8, 14, 18, 20]):
select(q, [5+6*i$i=0..35000])[];  # Alois P. Heinz, Jun 16 2025

Select[Prime[Range[20000]], AllTrue[#+{8, 14, 18,20}, PrimeQ]&] (* Stefano Spezia, Jun 18 2025 *)

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A191456 Primes p such that the polynomial x^2+x+p generates only primes for x=1..9.

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A172456 Primes p such that (p, p+2, p+6, p+12, p+14, p+20) is a prime sextuple.

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A291635 Number of ways to write n as x^2 + y^2 + z^2 + w^2 with x,y,z,w nonnegative integers such that p = x + 2*y + 5*z, p - 2, p + 4 and p + 10 are all prime.

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A341267 Gaps between first elements of quadruple primes of the form {p, p+2, p+6, p+12}.

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A385035 Primes p such that p + 8, p + 14, p + 18 and p + 20 are also primes.

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A291635 Number of ways to write n as x^2 + y^2 + z^2 + w^2 with x,y,z,w nonnegative integers such that p = x + 2y + 5z, p - 2, p + 4 and p + 10 are all prime.