A327637 a(n) is the number of integers j such that 1 <= j <= n and gcd(n,j) is a triangular number.

1, 1, 3, 2, 4, 4, 6, 4, 8, 5, 10, 7, 12, 6, 13, 8, 16, 10, 18, 9, 19, 10, 22, 14, 20, 12, 24, 13, 28, 19, 30, 16, 30, 16, 24, 19, 36, 18, 36, 18, 40, 25, 42, 20, 35, 22, 46, 28, 42, 24, 48, 24, 52, 30, 41, 25, 54, 28, 58, 32, 60, 30, 50, 32, 48, 41, 66, 32, 66, 30, 70, 37, 72, 36, 64

Offset

1,3

Comments

Moebius transform of A327629.

Links

Table of n, a(n) for n=1..75.

Formula

G.f.: Sum_{k>=1} phi(k) * (theta_2(x^(k/2)) / (2 * x^(k/8)) - 1), where phi() is the Euler totient function and theta_() is the Jacobi theta function.

a(n) = Sum_{d|n} A010054(n/d) * phi(d).

Mathematica

Table[Length[Select[Range[n], IntegerQ[(8 GCD[n, #] + 1)^(1/2)] &]], {n, 1, 75}]
Table[DivisorSum[n, Boole[IntegerQ[(8 n/# + 1)^(1/2)]] EulerPhi[#] &], {n, 1, 75}]
nmax = 75; CoefficientList[Series[Sum[EulerPhi[k] (EllipticTheta[2, 0, x^(k/2)]/(2 x^(k/8)) - 1), {k, 1, nmax}], {x, 0, nmax}], x] // Rest

Prog

(PARI) a(n) = sum(k=1, n, ispolygonal(gcd(k, n), 3)); \\ Michel Marcus, Sep 20 2019
(Magma) [#[j:j in [1..n]| IsSquare(8*Gcd(n, j)+1)]:n in [1..75]]; // Marius A. Burtea, Sep 20 2019

Crossrefs

Cf. A000010, A000217, A010054, A206369, A327629.

Sequence in context: A112199 A145815 A059851 * A366409 A345082 A047993

Adjacent sequences: A327634 A327635 A327636 * A327638 A327639 A327640

Keyword

nonn

Author

Ilya Gutkovskiy, Sep 20 2019

Status

approved