A349787 Number of ways to write n as x^2 + y^k + 2^a + 2^b, where x,y,a,b are nonnegative integers with x >= y and a >= b, and k is either 2 or 3.

2, 4, 6, 6, 8, 8, 8, 8, 11, 9, 11, 11, 12, 8, 9, 11, 15, 14, 16, 16, 17, 8, 10, 14, 15, 12, 16, 16, 12, 7, 11, 17, 22, 16, 17, 18, 17, 10, 16, 22, 23, 15, 17, 19, 17, 8, 15, 23, 19, 11, 20, 23, 17, 12, 17, 20, 20, 14, 18, 18, 13, 7, 12, 21, 23, 21, 25, 27, 26, 11, 17, 27, 25, 15, 22, 24, 14, 8, 17, 27, 29, 20, 29, 28

Offset

2,1

Comments

Conjecture 1: a(n) > 0 for all n >= 2.

Below is our weaker version of Conjecture 1.

Conjecture 2: Each n = 2,3,... can be written as a sum of two perfect powers (including 0 and 1) and two powers of 2 (including 2^0 = 1).

In contrast, R. Crocker proved in 2008 that there are infinitely many positive integers which cannot be written as a sum of two squares and two powers of 2.

Links

Zhi-Wei Sun, Table of n, a(n) for n = 2..10000

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

Dave Platt and Tim Trudgian, On the sum of two squares and at most two powers of 2, arXiv:1610.01672 [math.NT], 2016.

Example

a(2) = 2 with 2 = 0^2 + 0^2 + 2^0 + 2^0 = 0^2 + 0^3 + 2^0 + 2^0.
a(535903) > 0 since 535903 = 336^2 + 31^3 + 2^18 + 2^17 with 336 >= 31 and 18 >= 17.

Mathematica

PowQ[n_]:=PowQ[n]=IntegerQ[Log[2, n]];
tab={}; Do[r=0; Do[If[PowQ[n-x^2-y^k-2^a], r=r+1], {x, 0, Sqrt[n-2]}, {k, 2, 3}, {y, 0, Min[x, (n-2-x^2)^(1/k)]}, {a, 0, Log[2, n-x^2-y^k]-1}]; tab=Append[tab, r], {n, 2, 85}]; Print[tab]

Crossrefs

Cf. A000079, A000290, A000578, A001597, A002760, A273812, A303363, A303637, A349778.

Sequence in context: A114218 A133691 A111973 * A161655 A092517 A128558

Adjacent sequences: A349784 A349785 A349786 * A349788 A349789 A349790

Keyword

nonn

Author

Zhi-Wei Sun, Nov 30 2021

Status

approved