login
A262985
Number of ordered ways to write n as 2^x + phi(y^2) + z*(z+1)/2 with x, y and z positive integers, where phi(.) is Euler's totient function given by A000010.
2
0, 0, 0, 1, 1, 2, 2, 1, 3, 2, 5, 2, 5, 2, 5, 4, 4, 4, 5, 7, 3, 3, 5, 5, 8, 4, 5, 3, 5, 4, 8, 4, 3, 6, 5, 2, 9, 6, 8, 4, 5, 5, 8, 6, 8, 8, 4, 6, 8, 10, 7, 6, 7, 8, 9, 6, 7, 7, 12, 5, 9, 8, 6, 7, 12, 5, 9, 6, 9, 6, 11, 9, 11, 5, 6, 10, 8, 7, 9, 11, 5, 7, 7, 8, 7, 9, 8, 8, 9, 6, 7, 9, 7, 10, 9, 4, 6, 6, 7, 9
OFFSET
1,6
COMMENTS
Conjecture: a(n) > 0 for all n > 3.
We have verified this for n up to 1.3*10^8.
EXAMPLE
a(4) = 1 since 4 = 2 + phi(1^2) + 1*2/2.
a(5) = 1 since 5 = 2 + phi(2^2) + 1*2/2.
a(8) = 1 since 8 = 2^2 + phi(1^2) + 2*3/2.
a(36) = 2 since 36 = 2 + phi(3^2) + 7*8/2 = 2^5 + phi(1^2) + 2*3/2.
MATHEMATICA
f[n_]:=EulerPhi[n^2]
TQ[n_]:=n>0&&IntegerQ[Sqrt[8n+1]]
Do[r=0; Do[If[f[x]>=n, Goto[aa]]; Do[If[TQ[n-f[x]-2^y], r=r+1], {y, 1, Log[2, n-f[x]]}]; Label[aa]; Continue, {x, 1, n}]; Print[n, " ", r]; Continue, {n, 1, 100}]
KEYWORD
nonn
AUTHOR
Zhi-Wei Sun, Oct 06 2015
STATUS
approved