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A305444
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a(n) = Product_{p is odd and prime and divisor of n} (p - 2).
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3
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1, 1, 1, 1, 3, 1, 5, 1, 1, 3, 9, 1, 11, 5, 3, 1, 15, 1, 17, 3, 5, 9, 21, 1, 3, 11, 1, 5, 27, 3, 29, 1, 9, 15, 15, 1, 35, 17, 11, 3, 39, 5, 41, 9, 3, 21, 45, 1, 5, 3, 15, 11, 51, 1, 27, 5, 17, 27, 57, 3, 59, 29, 5, 1, 33, 9, 65, 15, 21, 15, 69, 1, 71, 35, 3, 17
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OFFSET
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1,5
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COMMENTS
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Denominator of c_n = Product_{odd p| n} (p-1)/(p-2). Numerator is A173557. [Yamasaki and Yamasaki]. - N. J. A. Sloane, Jan 19 2020
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LINKS
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FORMULA
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Sum_{k=1..n} a(k) ~ c * n^2, where c = (2/3) * Product_{p prime} (1 - 3/(p*(p+1))) = 0.1950799046... . - Amiram Eldar, Nov 12 2022
a(n) = abs( Sum_{d divides n, d odd} mobius(d) * phi(d) ). - Peter Bala, Feb 01 2024
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MAPLE
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A305444 := proc(n) mul(d - 2, d = numtheory[factorset](n) minus {2}) end proc:
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MATHEMATICA
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a[n_] := If[n == 1, 1, Times @@ (DeleteCases[FactorInteger[n][[All, 1]], 2] - 2)];
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PROG
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(PARI) a(n)={my(f=factor(n>>valuation(n, 2))[, 1]); prod(i=1, #f, f[i]-2)} \\ Andrew Howroyd, Aug 12 2018
(Python)
from math import prod
from sympy import primefactors
def A305444(n): return prod(p-2 for p in primefactors(n>>(~n&n-1).bit_length())) # Chai Wah Wu, Sep 08 2023
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CROSSREFS
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KEYWORD
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nonn,easy,mult,changed
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AUTHOR
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STATUS
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approved
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