

A308734


Number of ordered ways to write n as (2^a*3^b)^2 + (2^c*5^d)^2 + x^2 + y^2, where a,b,c,d,x,y are nonnegative integers with x <= y.


2



0, 1, 1, 1, 2, 3, 3, 1, 3, 5, 2, 3, 4, 4, 5, 1, 4, 8, 4, 4, 8, 8, 4, 3, 8, 7, 7, 6, 5, 13, 6, 1, 10, 11, 7, 7, 10, 9, 9, 5, 7, 18, 7, 5, 14, 11, 6, 3, 10, 11, 9, 8, 7, 15, 9, 4, 14, 12, 5, 10, 9, 10, 11, 1, 11, 19, 10, 6, 17, 21, 6, 8, 14, 12, 13, 7, 14, 21, 7, 4
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OFFSET

1,5


COMMENTS

Foursquare Conjecture: a(n) > 0 for all n > 1.
This is much stronger than Lagrange's foursquare theorem. We have verified a(n) > 0 for all n = 2..10^9.
Note that 16265031 cannot be written as (2^a*3^b)^2 + (2^c*3^d)^2 + x^2 + y^2 with a,b,c,d,x,y nonnegative integers.
a(n) > 0 for 1 < n <= 10^10.  Giovanni Resta, Jun 28 2019
I promise to offer 2500 US dollars as the prize for the first correct proof of the Foursquare Conjecture.  ZhiWei Sun, Jul 09 2019


LINKS

ZhiWei Sun, Table of n, a(n) for n = 1..10000
ZhiWei Sun, Refining Lagrange's foursquare theorem, J. Number Theory 175 (2017), 167190.
ZhiWei Sun, Restricted sums of four squares, Int. J. Number Theory 15 (2019), 18631893.
ZhiWei Sun, Various Refinements of Lagrange's FourSquare Theorem, Westlake Number Theory Symposium (Nanjing University, China, 2020).


EXAMPLE

a(2^(2k+1)) = 1 with 2^(2k+1) = (2^k*3^0)^2 + (2^k*5^0)^2 + 0^2 + 0^2.
a(2^(2k+2)) = 1 with 2^(2k+2) = (2^k*3^0)^2 + (2^k*5^0)^2 + (2^k)^2 + (2^k)^2.
a(3) = 1 with 3 = (2^0*3^0)^2 + (2^0*5^0)^2 + 0^2 + 1^2.
a(5) = 2 with 5 = (2^0*3^0)^2 + (2^1*5^0)^2 + 0^2 + 0^2 = (2^1*3^0)^2 + (2^0*5^0)^2 + 0^2 + 0^2.
a(11) = 2 with 11 = (2^0*3^0)^2 + (2^0*5^0)^2 + 0^2 + 3^2 = (2^0*3^1)^2 + (2^0*5^0)^2 + 0^2 + 1^2.


MATHEMATICA

SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]];
tab={}; Do[r=0; Do[If[SQ[n4^a*9^b4^c*25^dx^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[25, (n4^a*9^b)/4^c]}, {x, 0, Sqrt[(n4^a*9^b4^c*25^d)/2]}]; tab=Append[tab, r], {n, 1, 80}]; Print[tab]


CROSSREFS

Cf. A000079, A000118, A000290, A000244, A000351, A271518, A281976, A303656, A308566, A308584, A308621, A308623, A308640, A308641, A308644, A308656, A308661, A308662.
Sequence in context: A279813 A256909 A343533 * A279004 A172528 A087074
Adjacent sequences: A308731 A308732 A308733 * A308735 A308736 A308737


KEYWORD

nonn


AUTHOR

ZhiWei Sun, Jun 21 2019


STATUS

approved



