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.

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.

Links

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

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}]

Crossrefs

Cf. A000010, A000079, A000217, A000290, A002618, A262976, A262980, A262982.

Sequence in context: A276427 A129711 A030454 * A296786 A145141 A103360

Adjacent sequences: A262982 A262983 A262984 * A262986 A262987 A262988

Keyword

nonn

Author

Zhi-Wei Sun, Oct 06 2015

Status

approved