

A308934


Number of ways to write n as (2^a*3^b)^2 + (2^c*3^d)^2 + x^2 + 2*y^2, where a,b,c,d,x,y are nonnegative integers with 2^a*3^b >= 2^c*3^d.


1



0, 1, 1, 1, 2, 2, 1, 3, 2, 3, 5, 2, 4, 6, 1, 4, 5, 4, 7, 6, 7, 6, 4, 6, 4, 9, 7, 5, 10, 4, 4, 7, 4, 7, 10, 7, 8, 9, 4, 8, 10, 7, 10, 9, 7, 11, 5, 6, 11, 7, 10, 8, 11, 11, 5, 14, 6, 9, 13, 3, 13, 9, 6, 12, 7, 6, 11, 12, 12, 11, 10, 10, 10, 17, 9, 14, 14, 8, 10, 9, 14, 11, 16, 15, 13, 18, 6, 14, 17, 14, 22, 11, 12, 16, 7, 13, 11, 16, 19, 13
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OFFSET

1,5


COMMENTS

Conjecture 1: a(n) > 0 for all n > 1.
Conjecture 2: Any integer n > 1 can be written as (2^a*3^b)^2 + (2^c*5^d)^2 + x^2 + 2*y^2 with a,b,c,d,x,y nonnegative integers.
These two conjectures are similar to the Foursquare Conjecture in A308734. We have verified Conjectures 1 and 2 for n up to 2*10^9 and 10^9 respectively.


LINKS

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


EXAMPLE

a(3) = 1 with 3 = 1^2 + 1^2 + 1^2 + 2*0^2.
a(7) = 1 with 7 = 2^2 + 1^2 + 0^2 + 2*1^2.
a(15) = 1 with 15 = 3^2 + 2^2 + 0^2 + 2*1^2.


MATHEMATICA

SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]];
tab={}; Do[r=0; Do[If[SQ[n4^a*9^b4^c*9^d2x^2], r=r+1], {a, 0, Log[4, n]}, {b, 0, Ceiling[Log[9, n/4^a]]1},
{c, 0, Log[4, n4^a*9^b]}, {d, 0, Log[9, Min[4^(ac)*9^b, (n4^a*9^b)/4^c]]}, {x, 0, Sqrt[(n4^a*9^b4^c*9^d)/2]}]; tab=Append[tab, r], {n, 1, 100}]; Print[tab]


CROSSREFS

Cf. A000079, A000244, A000290, A000351, A275344, A308734.
Sequence in context: A289438 A008678 A159803 * A058741 A339491 A291712
Adjacent sequences: A308931 A308932 A308933 * A308935 A308936 A308937


KEYWORD

nonn


AUTHOR

ZhiWei Sun, Jul 01 2019


STATUS

approved



