A263992 Number of ordered ways to write n as x^2 + 2*y^2 + phi(z^2) (x >= 0, y >= 0 and z > 0) such that y or z has the form p-1 with p prime, where phi(.) is Euler's totient function.

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

Offset

1,2

Comments

Conjecture: a(n) > 0 for all n > 0, and a(n) = 1 only for n = 1, 6, 7, 22, 3447.

This is similar to the conjecture in A262311, and we have verified it for n up to 10^6.

Links

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

Zhi-Wei Sun, Some mysterious representations of integers, a message to Number Theory Mailing List, Oct. 25, 2015.

Example

a(1) = 1 since 1 = 0^2 + 2*0^2 + phi(1^2) with 1 + 1 = 2 prime.
a(6) = 1 since 6 = 2^2 + 2*0^2 + phi(2^2) with 2 + 1 = 3 prime.
a(7) = 1 since 7 = 2^2 + 2*1^2 + phi(1^2) with 1 + 1 = 2 prime.
a(22) = 1 since 22 = 0^2 + 2*1^2 + phi(5^2) with 1 + 1 = 2 prime.
a(3447) = 1 since 3447 = 42^2 + 2*29^2 + phi(1^2) with 1 + 1 = 2 prime.

Mathematica

phi[n_]:=phi[n]=EulerPhi[n]
SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]]
Do[r=0; Do[If[n-z*phi[z]<0, Goto[aa]]; Do[If[SQ[n-z*phi[z]-2y^2]&&(PrimeQ[y+1]||PrimeQ[z+1]), r=r+1], {y, 0, Sqrt[(n-z*phi[z])/2]}]; Label[aa]; Continue, {z, 1, n}]; Print[n, " ", r]; Continue, {n, 1, 100}]

Crossrefs

Cf. A000010, A000290, A006093, A262311, A263998.

Sequence in context: A057155 A037812 A037200 * A180174 A336013 A172363

Adjacent sequences: A263989 A263990 A263991 * A263993 A263994 A263995

Keyword

nonn

Author

Zhi-Wei Sun, Oct 31 2015

Status

approved