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