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A229166 Number of ordered ways to write n = x*(x+1)/2 + y with y*(y+1)/2 + 1 prime, where x and y are nonnegative integers. 5
1, 1, 1, 3, 1, 1, 3, 2, 2, 2, 3, 2, 3, 3, 1, 2, 3, 3, 3, 2, 2, 5, 3, 2, 2, 4, 2, 2, 4, 2, 2, 2, 3, 1, 3, 2, 3, 2, 2, 4, 3, 1, 3, 5, 2, 3, 4, 5, 2, 4, 2, 3, 3, 2, 3, 5, 4, 2, 4, 1, 4, 3, 5, 4, 3, 5, 3, 4, 3, 3, 6, 4, 2, 5, 4, 3, 4, 5, 5, 2, 4, 4, 2, 3, 6, 4, 2, 3, 5, 4, 3, 5, 1, 4, 3, 6, 3, 5, 7, 3 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,4

COMMENTS

Conjecture: a(n) > 0 for all n > 0. Moreover, if n > 0 is not among 1, 3, 60, then there are positive integers x and y with x*(x+1)/2 + y = n such that y*(y+1)/2 + 1 is prime.

REFERENCES

Zhi-Wei Sun, On sums of primes and triangular numbers, J. Comb. Number Theory 1(2009), no.1, 65-76.

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(6) = 1 since 6 = 2*3/2 + 3 with 3*4/2 + 1 = 7 prime.

a(60) = 1 since 60 = 0*1/2 + 60 with 60*61/2 + 1 = 1831 prime.

MATHEMATICA

T[n_]:=n(n+1)/2

a[n_]:=Sum[If[PrimeQ[T[n-T[i]]+1], 1, 0], {i, 0, (Sqrt[8n+1]-1)/2}]

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

CROSSREFS

Cf. A000040, A000217, A132399, A208244, A230121, A230252.

Sequence in context: A077196 A023142 A225335 * A143159 A324079 A184831

Adjacent sequences:  A229163 A229164 A229165 * A229167 A229168 A229169

KEYWORD

nonn

AUTHOR

Zhi-Wei Sun, Oct 15 2013

STATUS

approved

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Last modified October 17 13:57 EDT 2019. Contains 328113 sequences. (Running on oeis4.)