A262979 - OEIS

%I #9 Oct 06 2015 04:00:28

%S 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,

%T 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,

%U 5,6,7,5,4,3,5,5,5,5,6,4,3,5,8,7,6,4,5,5,8,8,5,5

%N 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.

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

%C (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.

%H Zhi-Wei Sun, <a href="/A262979/b262979.txt">Table of n, a(n) for n = 1..10000</a>

%e a(5) = 1 since 5 = 1^4 + phi(2^2) + (-1)*(3*(-1)-1)/2.

%e 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.

%e a(16) = 2 since 16 = 0^4 + phi(1^2) + (-3)*(3*(-3)-1)/2

%e = 1^4 + phi(4^2) + (-2)*(3*(-2)-1)/2.

%t f[n_]:=EulerPhi[n^2]

%t PenQ[n_]:=IntegerQ[Sqrt[24n+1]]

%t 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}]

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

%K nonn

%O 1,2

%A _Zhi-Wei Sun_, Oct 06 2015