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A353909
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a(n) is the alternating sum of the sequence gcd(n, k^2), 1 <= k <= n.
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1
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-1, 1, -3, 6, -5, 5, -7, 20, -9, 9, -11, 30, -13, 13, -15, 72, -17, 33, -19, 54, -21, 21, -23, 100, -25, 25, -27, 78, -29, 45, -31, 208, -33, 33, -35, 198, -37, 37, -39, 180, -41, 65, -43, 126, -45, 45, -47, 360, -49, 145, -51, 150, -53, 153, -55, 260, -57, 57, -59
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OFFSET
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1,3
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LINKS
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FORMULA
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a(n) = Sum_{i=1..n} (-1)^i*gcd(n, i^2).
a(n) = -n if n is odd.
a(n) = n * Sum_{d|n, d even} (phi(d) * sqrt(d/core(d)) / d), where phi = A000010, if n is even. - Darío Clavijo, Jan 13 2023
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MAPLE
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a:= n-> add((-1)^i*igcd(n, i^2), i=1..n):
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MATHEMATICA
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a[n_] := Sum[(-1)^i * GCD[n, i^2], {i, 1, n}]; Array[a, 100] (* Amiram Eldar, May 10 2022 *)
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PROG
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(PARI) a(n) = sum(i=1, n, (-1)^i*gcd(n, i^2)); \\ Michel Marcus, May 10 2022
(PARI) a(n) = {
if((n%2)==1, return(-n));
my(s=0);
fordivfactored(n, d,
if((d[1]%2)==0,
s+=eulerphi(d)*core(d, 1)[2]/d[1]));
s*n;
(Python)
from math import gcd
def a(n):
return -n if n%2==1 else sum((-1)**k*gcd(n, k*k) for k in range(1, n+1))
(Python)
from sympy import sqrt, divisors, totient
from sympy.ntheory.factor_ import core
def a(n):
return -n if n & 1 == 1 else int(n * sum(totient(d) * sqrt(d // core(d)) / d for d in divisors(n) if d & 1 == 0))
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CROSSREFS
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KEYWORD
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sign
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AUTHOR
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STATUS
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approved
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