A129634 -id:A129634 - cp's OEIS frontend

A130504 Number of k for which T(n) + T(k) is prime, with 0<=k<=n and triangular number T(n)=n(n+1)/2.

0, 1, 1, 1, 2, 0, 1, 3, 1, 1, 2, 1, 1, 4, 0, 1, 8, 2, 3, 4, 1, 3, 7, 3, 0, 4, 3, 3, 6, 2, 1, 6, 2, 3, 6, 2, 3, 7, 4, 2, 8, 2, 4, 7, 2, 3, 15, 5, 3, 6, 2, 5, 13, 5, 1, 6, 2, 3, 21, 3, 3, 14, 3, 6, 7, 2, 5, 15, 6, 3, 6, 5, 9, 15, 4, 3, 12, 3, 6, 18, 3, 7, 16, 4, 6, 7, 7, 5, 15, 1, 4, 17, 5, 6, 9, 7, 8, 18, 6, 5

Offset: 0

T. D. Noe, Jun 04 2007

nonn

It appears that a(n)=0 for n=0,5,14,24 only. See A129634 for the least k.

			a(4)=2 because 10+1 and 10+3 are prime; a(7)=3 because 28+1, 28+3 and 28+15 are primes.

T. D. Noe, Table of n, a(n) for n = 0..10000

Cf. A069004 (for square numbers).

nn=100; tri=Range[0,nn]Range[nn+1]/2; Table[cnt=0; Do[If[PrimeQ[tri[[k]]+tri[[n]]], cnt++ ], {k,n}]; cnt, {n,Length[tri]}]

A210646 Primes which are the sum of two numbers of the form k*(k+1)^2/2.

Original entry on oeis.org

2, 11, 59, 149, 197, 233, 607, 829, 929, 1283, 1619, 1879, 2459, 2917, 3251, 3299, 3359, 3947, 4523, 5821, 5843, 5869, 6043, 6143, 6269, 6833, 7229, 8573, 8597, 9479, 9619, 11699, 11783, 11789, 12379, 14639, 15881, 16477, 18869, 19121, 20849, 21149, 21617

Offset: 1

Gerasimov Sergey, Mar 26 2012

nonn

			149 is in the sequence because 149 is prime and 149 = 2*(2+1)^2/2 + 6*(6+1)^2/2.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A006002, A129634, A210647.

f[n_] := n (n + 1)^2/2; t = Table[f[n], {n, 0, 40}]; Select[Union[Flatten[Outer[Plus, t, t]]], # < t[[-1]] && PrimeQ[#] &] (* T. D. Noe, Apr 03 2012 *)

Terms checked by R. J. Mathar, Mar 28 2012

A210647 Least nonnegative m such that k(n) + k(m) is prime, where k(n) = n*(n+1)^2/2.

Original entry on oeis.org

0, 1, 22, 2, 142, 1, 2, 10, 22, 1, 34, 10, 2, 37, 46, 6, 10, 1, 6, 46, 46, 1, 10, 106, 6, 1, 58, 2, 22, 7, 2, 58, 94, 3, 22, 10, 2, 1, 22, 2, 10, 16, 6, 82, 118, 4, 82, 10, 18, 1, 10, 2, 22, 1, 2, 10, 10, 4, 22, 58, 2, 19, 10, 2, 46, 1, 10, 70, 82, 3, 22, 34

Offset: 1

Gerasimov Sergey, Mar 27 2012

nonn

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A006002, A129634, A210646.

f[n_] := n (n + 1)^2/2; Table[k = 0; While[! PrimeQ[f[n] + f[k]], k++]; k, {n, 100}] (* T. D. Noe, Apr 03 2012 *)

a(n)=my(K=n*(n+1)^2/2,m);while(!isprime(K+m*(m+1)^2/2),m++);m \\ Charles R Greathouse IV, Aug 03 2012

Corrected by R. J. Mathar, Mar 31 2012

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A130504 Number of k for which T(n) + T(k) is prime, with 0<=k<=n and triangular number T(n)=n(n+1)/2.

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A210646 Primes which are the sum of two numbers of the form k*(k+1)^2/2.

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A210647 Least nonnegative m such that k(n) + k(m) is prime, where k(n) = n*(n+1)^2/2.

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