

A262954


Number of ordered pairs (x,y) with x >= 0 and y > 0 such that n  x^4  y*(y+1)/2 has the form z^2 or 8*z^2.


15



1, 2, 2, 2, 2, 2, 3, 2, 1, 4, 5, 3, 1, 2, 4, 3, 3, 3, 5, 4, 2, 4, 5, 2, 3, 6, 4, 3, 4, 5, 5, 4, 3, 2, 5, 4, 7, 7, 3, 4, 3, 4, 2, 4, 6, 6, 6, 2, 2, 2, 4, 5, 9, 5, 4, 5, 2, 3, 2, 5, 5, 5, 2, 4, 5, 3, 4, 5, 4, 5, 7, 3, 3, 3, 6, 3, 4, 4, 5, 6, 3, 7, 7, 3, 4, 8, 7, 7, 1, 3, 9, 8, 6
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OFFSET

1,2


COMMENTS

Conjecture: (i) a(n) > 0 for all n > 0, and a(n) = 1 only for n = 1, 9, 13, 89, 449, 524, 1049, 2164, 14969, 51334.
(ii) For any positive integer n, there are integers x and y > 0 such that n  x^4  T(y) has the form T(z) or 4*T(z), where T(k) refers to the triangular number k*(k+1)/2.
(iii) For every n = 1,2,3,... there are integers x and y > 0 such that n  x^4  T(y) has the form T(z) or 2*z^2.
(iv) For {c,d} = {1,2} and n > 0, there are integers x and y > 0 such that n  2*x^4  T(y) has the form c*T(z) or d*z^2.
(v) For each n = 1,2,3,... there are integers x and y > 0 such that n  4*x^4  T(y) has the form 2*T(z) or z^2.
See also A262941, A262944, A262945, A262954, A262955 and A262956 for similar conjectures.


LINKS

ZhiWei Sun, Table of n, a(n) for n = 1..10000
ZhiWei Sun, Mixed sums of squares and triangular numbers, Acta Arith. 127(2007), 103113.


EXAMPLE

a(1) = 1 since 1 = 0^4 + 1*2/2 + 0^2.
a(9) = 1 since 9 = 0^4 + 1*2/2 + 8*1^2.
a(13) = 1 since 13 = 1^4 + 2*3/2 + 3^2.
a(89) = 1 since 89 = 2^4 + 1*2/2 + 8*3^2.
a(449) = 1 since 449 = 0^4 + 22*23/2 + 14^2.
a(524) = 1 since 524 = 3^4 + 29*30/2 + 8*1^2.
a(1049) = 1 since 1049 = 5^4 + 16*17/2 + 8*6^2.
a(2164) = 1 since 2164 = 1^4 + 34*35/2 + 8*14^2.
a(14969) = 1 since 14969 = 8^4 + 145*146/2 + 8*6^2.
a(51334) = 1 since 51334 = 5^4 + 313*314/2 + 8*14^2.


MATHEMATICA

SQ[n_]:=IntegerQ[Sqrt[n]]IntegerQ[Sqrt[n/8]]
Do[r=0; Do[If[SQ[nx^4y(y+1)/2], r=r+1], {x, 0, n^(1/4)}, {y, 1, (Sqrt[8(nx^4)+1]1)/2}]; Print[n, " ", r]; Continue, {n, 1, 100}]


CROSSREFS

Cf. A000217, A000290, A000583, A262941, A262944, A262945, A262954, A262955, A262956.
Sequence in context: A068323 A054990 A046921 * A262813 A188794 A161966
Adjacent sequences: A262951 A262952 A262953 * A262955 A262956 A262957


KEYWORD

nonn


AUTHOR

ZhiWei Sun, Oct 05 2015


STATUS

approved



