A230514 -id:A230514 - cp's OEIS frontend

A230515 Numbers n such that n*(n+1)-1 is a Sophie Germain prime.

2, 3, 5, 6, 9, 11, 15, 20, 38, 39, 45, 48, 50, 54, 59, 93, 126, 131, 144, 149, 153, 174, 176, 218, 231, 236, 240, 246, 248, 263, 285, 306, 309, 330, 335, 374, 380, 395, 396, 401, 419, 423, 449, 455, 468, 471, 474, 495, 501, 506, 549, 551, 560, 588

Offset: 1

Zhi-Wei Sun, Oct 21 2013

nonn

This sequence is interesting because of the conjecture associated with A230514.

			a(1) = 2 since 2*3 - 1 = 5 is a Sophie Germain prime.
a(2) = 3 since 3*4 - 1 = 11 is a Sophie Germain prime.
a(3) = 5 since 5*6 - 1 = 29 is a Sophie Germain prime but 4*5 - 1 = 19 is not.

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000

Cf. A000040, A005384, A230514.

Subsequence of A045546.

[n: n in [1..600] | IsPrime(n*(n+1)-1) and IsPrime(2*n*(n+1)-1)]; // Bruno Berselli, Oct 22 2013

q[n_]:=PrimeQ[n(n+1)-1]&&PrimeQ[2n(n+1)-1]
m=0
Do[If[q[n],m=m+1;Print[m," ",n]],{n,1,506}]
Select[Range[600],AllTrue[{#^2+#-1,2#^2+2#-1},PrimeQ]&] (* Harvey P. Dale, Dec 02 2021 *)

A230516 Number of ways to write n = a + b + c with 0 < a <= b <= c such that {a^2+a-1, a^2+a+1}, {b^2+b-1, b^2+b+1}, {c^2+c-1, c^2+c+1} are twin prime pairs.

Original entry on oeis.org

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

Offset: 1

Zhi-Wei Sun, Oct 22 2013

nonn

Conjecture: a(n) > 0 for all n > 5.
Conjecture verified for n up to 10^9. - Mauro Fiorentini, Sep 22 2023
This implies that there are infinitely many twin prime pairs of the form {x^2 + x - 1, x^2 + x + 1}.
See also A230514 for a similar conjecture.

			a(8) = 1 since 8 = 2 + 3 + 3, and {2*3 - 1, 2*3 + 1} = {5, 7} and {3*4 - 1, 3*4 + 1} = {11, 13} are twin prime pairs.
a(39) = 1 since 39 = 3 + 15 + 21, and {3*4 - 1, 3*4 + 1} = {11, 13}, {15*16 - 1, 15*16 + 1} = {239, 241}, {21*22 - 1, 21*22 + 1} = {461, 463} are twin prime pairs.

Zhi-Wei Sun, Table of n, a(n) for n = 1..5000
Zhi-Wei Sun, Conjectures involving primes and quadratic forms, preprint, arXiv:1211.1588 [math.NT], 2012-2017.

Cf. A001359, A006512, A088485, A227923, A230219, A230252, A230351, A230514.

pp[n_]:=PrimeQ[n(n+1)-1]&&PrimeQ[n(n+1)+1]
a[n_]:=Sum[If[pp[i]&&pp[j]&&pp[n-i-j],1,0],{i,1,n/3},{j,i,(n-i)/2}]
Table[a[n],{n,1,100}]

cp's OEIS Frontend

A230515 Numbers n such that n*(n+1)-1 is a Sophie Germain prime.

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