A262979 Number of ordered ways to write n as x^4 + phi(y^2) + z*(3*z-1)/2 with x >= 0 and y > 0, where phi(.) is Euler's totient function given by A000010.

1, 3, 4, 3, 1, 2, 4, 5, 5, 3, 2, 2, 5, 6, 3, 2, 4, 5, 4, 4, 4, 5, 5, 7, 5, 2, 4, 6, 6, 3, 2, 4, 6, 5, 5, 4, 5, 4, 4, 4, 5, 7, 9, 6, 3, 4, 6, 9, 5, 6, 2, 4, 7, 6, 8, 6, 6, 8, 7, 7, 4, 4, 8, 6, 4, 4, 3, 5, 5, 6, 7, 5, 4, 3, 5, 5, 5, 5, 6, 4, 3, 5, 8, 7, 6, 4, 5, 5, 8, 8, 5, 5

Offset: 1

Author

Zhi-Wei Sun, Oct 06 2015

Keywords

• nonn

Comments

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

(ii) Any positive integer n can be written as x^4 + phi(y^2) + pi(z^2) (or x^4 + pi(y^2) + pi(z^2)) with y > 0 and z > 0, where pi(m) denotes the number of primes not exceeding m.

Examples

			a(5) = 1 since 5 = 1^4 + phi(2^2) + (-1)*(3*(-1)-1)/2.
a(6) = 2 since 6 = 0^4 + phi(1^2) + 2*(3*2-1)/2 = 0^4 + phi(3^2) + 0*(3*0-1)/2.
a(16) = 2 since 16 = 0^4 + phi(1^2) + (-3)*(3*(-3)-1)/2
= 1^4 + phi(4^2) + (-2)*(3*(-2)-1)/2.
		

Links

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

Crossrefs

Cf. A000010, A000290, A000583, A000720, A001318, A002618, A262311, A262746, A262781, A262887, A262941, A262976.

Programs

Mathematica

f[n_]:=EulerPhi[n^2]
PenQ[n_]:=IntegerQ[Sqrt[24n+1]]
Do[r=0;Do[If[f[x]>n,Goto[aa]];Do[If[PenQ[n-f[x]-y^4],r=r+1],{y,0,(n-f[x])^(1/4)}];Label[aa];Continue,{x,1,n}];Print[n," ",r];Continue,{n,1,100}]