

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

ZhiWei Sun, On sums of primes and triangular numbers, J. Comb. Number Theory 1(2009), no.1, 6576.


LINKS

ZhiWei Sun, Table of n, a(n) for n = 1..10000
ZhiWei 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[nT[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

ZhiWei Sun, Oct 15 2013


STATUS

approved



