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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

%I

%S 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,

%T 13,28,19,30,16,30,16,24,19,36,18,36,18,40,25,42,20,35,22,46,28,42,24,

%U 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

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

%C Moebius transform of A327629.

%F 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.

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

%t Table[Length[Select[Range[n], IntegerQ[(8 GCD[n, #] + 1)^(1/2)] &]], {n, 1, 75}]

%t Table[DivisorSum[n, Boole[IntegerQ[(8 n/# + 1)^(1/2)]] EulerPhi[#] &], {n, 1, 75}]

%t 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

%o (PARI) a(n) = sum(k=1, n, ispolygonal(gcd(k,n), 3)); \\ _Michel Marcus_, Sep 20 2019

%o (MAGMA) [#[j:j in [1..n]| IsSquare(8*Gcd(n,j)+1)]:n in [1..75]]; // _Marius A. Burtea_, Sep 20 2019

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

%K nonn

%O 1,3

%A _Ilya Gutkovskiy_, Sep 20 2019

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Last modified May 7 10:18 EDT 2021. Contains 343650 sequences. (Running on oeis4.)