A280153 -id:A280153 - cp's OEIS frontend

A280356 Number of ways to write n as x^4 + y^3 + z^2 + 2^k, where x,y,z are nonnegative integers and k is a positive integer.

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

Offset: 1

Zhi-Wei Sun, Jan 01 2017

nonn

Conjecture: (i) a(n) > 0 for all n > 1, and a(n) = 1 only for n = 2, 23, 1135, 6415, 6471.
(ii)  If P(x,y) is one of the polynomials 3*x^4 + y^3 and x^6 + 3*y^2, then any positive integer n can be written as P(x,y) + z^2 + 2^k with x,y,z and k nonnegative integers.
We have verified that a(n) > 0 for all n = 2..2*10^7, and that part (ii) of the conjecture holds for all n = 1..10^7.
We also find finitely many polynomials of the form a*x^m + b*y^2 (including x^4 + y^2 and 10*x^5 + y^2) with a and b positive integers and m <= 5, for which it seems that any positive integer can be written as P(x,y) + z^2 + 2^k with x,y,z,k nonnegative integers.
See also A280153 for a similar conjecture involving powers of 4 or 8.
Qing-Hu Hou at Tianjin Univ. has verified that a(n) > 0 for all n = 2..10^9. In 2017, the author announced to offer US $234 as the prize for the first correct solution to his conjecture that a(n) > 0 for all n > 1. - Zhi-Wei Sun, Dec 30 2017

			a(2) = 1 since 2 = 0^4 + 0^3 + 0^2 + 2^1.
a(23) = 1 since 23 = 2^4 + 1^3 + 2^2 + 2^1.
a(1135) = 1 since 1135 = 0^4 + 7^3 + 28^2 + 2^3.
a(6415) = 1 since 6415 = 1^4 + 13^3 + 11^2 + 2^12.
a(6471) = 1 since 6471 = 1^4 + 13^3 + 57^2 + 2^10.

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
Zhi-Wei Sun, New conjectures on representations of integers (I), Nanjing Univ. J. Math. Biquarterly 34(2017), no. 2, 97-120.

Cf. A000079, A000290, A000578, A000583, A262827, A262956, A270533, A270559.

SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]];
In[2]:= Do[r=0;Do[If[SQ[n-2^k-x^4-y^3],r=r+1],{k,1,Log[2,n]},{x,0,(n-2^k)^(1/4)},{y,0,(n-2^k-x^4)^(1/3)}];Print[n," ",r];Continue,{n,1,80}]

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A280356 Number of ways to write n as x^4 + y^3 + z^2 + 2^k, where x,y,z are nonnegative integers and k is a positive integer.

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