

A227877


Number of ways to write n = x + y + z (x, y, z > 0) such that x*y and x*z are triangular numbers, and 6*y1 and 6*z+1 are both prime.


3



0, 0, 1, 0, 3, 2, 2, 3, 3, 7, 3, 6, 3, 3, 2, 3, 7, 6, 7, 5, 4, 5, 10, 2, 10, 4, 5, 2, 2, 9, 5, 9, 2, 4, 3, 4, 5, 7, 5, 11, 12, 5, 8, 11, 12, 5, 11, 3, 7, 11, 4, 10, 6, 2, 9, 11, 8, 7, 9, 8, 9, 4, 3, 4, 10, 6, 9, 15, 9, 17, 3, 3, 8, 12, 10, 5, 1, 7, 9, 16, 8, 17, 6, 8, 16, 6, 8, 8, 10, 1, 6, 4, 8, 5, 23, 11, 2, 9, 6, 14
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OFFSET

1,5


COMMENTS

Conjecture: a(n) > 0 for all n > 4.
For n = 4*k  1, we have n = (2k1) + k + k with (2k1)*k = 2k*(2k1)/2 a triangular number. For n = 4*k + 1, we have n = (2k+1) + k + k with (2k+1)*k = 2k*(2k+1)/2 a triangular number. For n = 4*k + 2, we have n = (2k+1) + k + (k+1), and (2k+1)*k = 2k*(2k+1)/2 and (2k+1)*(k+1) = (2k+1)(2k+2)/2 are both triangular numbers.
For n = 5*k, we have n = k + (2k1) + (2k+1), and k*(2k1) = 2k*(2k1)/2 and k*(2k+1) = 2k*(2k+1)/2 are both triangular numbers. For n = 5*k  2, we have n = k + (2k1) + (2k1) with k*(2k1) = 2k*(2k1)/2 a triangular number. For n = 5*k + 2, we have n = k + (2k+1) + (2k+1) with k*(2k+1) = 2k*(2k+1)/2 a triangular number.


LINKS

ZhiWei Sun, Table of n, a(n) for n = 1..10000
ZhiWei Sun, A new conjecture on triangular numbers, a message to Number Theory List, Oct. 25, 2013.


EXAMPLE

a(77) = 1 since 77 = 1 + 10 + 66, and 1*10 = 4*5/2 and 1*66 = 11*12/2 are triangular numbers, and 6*10  1 = 59 and 6*66 + 1 = 397 are both prime.
a(90) = 1 since 90 = 45 + 22 + 23, and 45*22 = 44*45/2 and 45*23 = 45*46/2 are triangular numbers, and 6*22  1 = 131 and 6*23 + 1 = 139 are both prime.


MATHEMATICA

TQ[n_]:=IntegerQ[Sqrt[8n+1]]
a[n_]:=Sum[If[PrimeQ[6j1]&&PrimeQ[6(nij)+1]&&TQ[i*j]&&TQ[i(nij)], 1, 0], {i, 1, n2}, {j, 1, n1i}]
Table[a[n], {n, 1, 100}]


CROSSREFS

Cf. A000040, A000217, A132399, A229166, A230121, A230451, A230596.
Sequence in context: A032536 A115061 A217834 * A225867 A046822 A129001
Adjacent sequences: A227874 A227875 A227876 * A227878 A227879 A227880


KEYWORD

nonn


AUTHOR

ZhiWei Sun, Oct 25 2013


STATUS

approved



