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A230254 Number of ways to write n = p + q with p and (p+1)*q/2 + 1 both prime. 4
0, 0, 0, 1, 1, 2, 1, 2, 2, 2, 3, 2, 1, 4, 1, 2, 5, 2, 3, 2, 3, 4, 4, 3, 4, 4, 2, 2, 8, 1, 6, 6, 2, 3, 2, 3, 5, 5, 5, 1, 5, 3, 7, 5, 1, 7, 10, 1, 3, 4, 8, 5, 3, 3, 3, 5, 8, 4, 10, 2, 9, 3, 3, 4, 7, 5, 9, 5, 4, 3, 15, 4, 12, 7, 4, 5, 9, 3, 11, 4, 6, 5, 9, 5, 6, 12, 6, 5, 8, 1, 4, 8, 5, 13, 9, 2, 6, 5, 8, 4 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,6

COMMENTS

Conjecture: a(n) > 0 for all n > 3.

We have verified this for n up to 10^8.

We also have some similar conjectures, for example, any integer n > 3 not equal to 17 or 66 can be written as p + q with p and (p+1)*q/2 - 1 both prime.

LINKS

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

Zhi-Wei Sun, Conjectures involving primes and quadratic forms, preprint, arXiv:1211.1588.

EXAMPLE

a(15) = 1 since 15 = 5 + 10 with 5 and (5+1)*10/2+1 = 31 both prime.

a(30) = 1 since 30 = 2 + 28 with 2 and (2+1)*28/2+1 = 43 both prime.

MATHEMATICA

a[n_]:=Sum[If[PrimeQ[(Prime[i]+1)(n-Prime[i])/2+1], 1, 0], {i, 1, PrimePi[n-1]}]

Table[a[n], {n, 1, 100}]

CROSSREFS

Cf. A000040, A002375, A219864, A230252, A227908, A227909, A230241.

Sequence in context: A057368 A192394 A085033 * A299484 A327007 A336767

Adjacent sequences:  A230251 A230252 A230253 * A230255 A230256 A230257

KEYWORD

nonn

AUTHOR

Zhi-Wei Sun, Oct 14 2013

STATUS

approved

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Last modified August 11 09:47 EDT 2020. Contains 336423 sequences. (Running on oeis4.)