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A303637 Number of ways to write n as x^2 + y^2 + 2^z + 5*2^w, where x,y,z,w are nonnegative integers with x <= y. 20
0, 0, 0, 0, 0, 1, 2, 2, 2, 2, 4, 3, 4, 5, 5, 5, 4, 4, 5, 4, 5, 9, 8, 6, 6, 9, 7, 6, 8, 8, 10, 8, 4, 8, 5, 7, 9, 12, 9, 6, 10, 9, 11, 10, 8, 16, 12, 8, 9, 12, 9, 11, 12, 11, 9, 10, 12, 14, 10, 10 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,7

COMMENTS

Conjecture: a(n) > 0 for all n > 5.

This has been verified for n up to 5*10^9. Note that 321256731 cannot be written as x^2 + (2*y)^2 + 2^z + 5*2^w with x,y,z,w nonnegative integers.

In contrast, Crocker 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.

570143 cannot be written as x^2 + y^2 + 2^z + 3*2^w with x,y,z,w nonnegative integers, and 2284095 cannot be written as x^2 + y^2 + 2^z + 7*2^w with x,y,z,w nonnegative integers.

REFERENCES

R. C. Crocker, On the sum of two squares and two powers of k, Colloq. Math. 112(2008), 235-267.

LINKS

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000

Zhi-Wei Sun, Refining Lagrange's four-square theorem, J. Number Theory 175(2017), 167-190.

Zhi-Wei Sun, New conjectures on representations of integers (I), Nanjing Univ. J. Math. Biquarterly 34(2017), no. 2, 97-120.

Zhi-Wei Sun, Restricted sums of four squares, arXiv:1701.05868 [math.NT], 2017-2018.

EXAMPLE

a(6) = 1 with 6 = 0^2 + 0^2 + 2^0 + 5*2^0.

a(8) = 2 with 8 = 1^2 + 1^2 + 2^0 + 5*2^0 = 0^2 + 1^2 + 2^1 + 5*2^0.

a(9) = 2 with 9 = 1^2 + 1^2 + 2^1 + 5*2^0 = 0^2 + 0^2 + 2^2 + 5*2^0.

a(10) = 2 with 10 = 0^2 + 2^2 + 2^0 + 5*2^0 = 0^2 + 1^2 + 2^2 + 5*2^0.

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[n-5*2^k-2^m], Do[If[SQ[n-5*2^k-2^m-x^2], r=r+1], {x, 0, Sqrt[(n-5*2^k-2^m)/2]}]], {k, 0, Log[2, n/5]}, {m, 0, If[n/5==2^k, -1, Log[2, n-5*2^k]]}]; tab=Append[tab, r], {n, 1, 60}]; Print[tab]

CROSSREFS

Cf. A000079, A000290, A001481, A273812, A302982, A302984, A303233, A303234, A303338, A303363, A303389, A303393, A303399, A303428, A303401, A303432, A303434, A303539, A303540, A303541, A303543, A303601, A303639, A303656.

Sequence in context: A097576 A029250 A110884 * A155904 A241514 A125913

Adjacent sequences:  A303634 A303635 A303636 * A303638 A303639 A303640

KEYWORD

nonn

AUTHOR

Zhi-Wei Sun, Apr 27 2018

STATUS

approved

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Last modified December 12 17:59 EST 2019. Contains 329960 sequences. (Running on oeis4.)