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A327637 a(n) is the number of integers j such that 1 <= j <= n and gcd(n,j) is a triangular number. 0
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 (list; graph; refs; listen; history; text; internal format)
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 * A047993 A033177 A175512

Adjacent sequences:  A327634 A327635 A327636 * A327638 A327639 A327640

KEYWORD

nonn

AUTHOR

Ilya Gutkovskiy, Sep 20 2019

STATUS

approved

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Last modified April 17 14:32 EDT 2021. Contains 343063 sequences. (Running on oeis4.)