

A303363


Number of ways to write n as a*(a+1)/2 + b*(b+1)/2 + 2^c + 2^d, where a,b,c,d are nonnegative integers with a <= b, c <= d and 2c.


32



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

1,3


COMMENTS

Conjecture: a(n) > 0 for all n > 1.
This is stronger than the author's conjecture in A303233. I have verified a(n) > 0 for all n = 2..10^9.
In contrast, Corcker proved in 2008 that there are infinitely many positive integers not representable as the sum of two squares and at most two powers of 2.


LINKS

ZhiWei Sun, Table of n, a(n) for n = 1..10000
R. C. Crocker, On the sum of two squares and two powers of k, Colloq. Math. 112(2008), 235267.
ZhiWei Sun, Refining Lagrange's foursquare theorem, J. Number Theory 175(2017), 167190.
ZhiWei Sun, New conjectures on representations of integers (I), Nanjing Univ. J. Math. Biquarterly 34(2017), no. 2, 97120.
ZhiWei Sun, Restricted sums of four squares, arXiv:1701.05868 [math.NT], 20172018.


EXAMPLE

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


MATHEMATICA

SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]];
f[n_]:=f[n]=FactorInteger[n];
g[n_]:=g[n]=Sum[Boole[Mod[Part[Part[f[n], i], 1], 4]==3&&Mod[Part[Part[f[n], i], 2], 2]==1], {i, 1, Length[f[n]]}]==0;
QQ[n_]:=QQ[n]=(n==0)(n>0&&g[n]);
tab={}; Do[r=0; Do[If[QQ[4(n4^j2^k)+1], Do[If[SQ[8(n4^j2^kx(x+1)/2)+1], r=r+1], {x, 0, (Sqrt[4(n4^j2^k)+1]1)/2}]], {j, 0, Log[4, n/2]}, {k, 2j, Log[2, n4^j]}]; tab=Append[tab, r], {n, 1, 70}]; Print[tab]


CROSSREFS

Cf. A000079, A000217, A271518, A273812, A281976, A299924, A299537, A299794, A300219, A300362, A300396, A300441, A301376, A301391, A301471, A301472, A302920, A302981, A302982, A302983, A302984, A302985, A303233, A303234, A303235, A303338, A303389.
Sequence in context: A105496 A167618 A211707 * A045796 A127684 A036012
Adjacent sequences: A303360 A303361 A303362 * A303364 A303365 A303366


KEYWORD

nonn


AUTHOR

ZhiWei Sun, Apr 22 2018


STATUS

approved



