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A356472
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Numerator of the average of gcd(i,n) for i = 1..n.
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2
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1, 3, 5, 2, 9, 5, 13, 5, 7, 27, 21, 10, 25, 39, 3, 3, 33, 7, 37, 18, 65, 63, 45, 25, 13, 75, 3, 26, 57, 9, 61, 7, 35, 99, 117, 14, 73, 111, 125, 9, 81, 65, 85, 42, 21, 135, 93, 5, 19, 39, 55, 50, 105, 9, 189, 65, 185, 171, 117, 6, 121, 183, 13, 4, 45, 105, 133, 66, 75, 351, 141, 35, 145, 219, 13, 74, 39, 125, 157
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OFFSET
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1,2
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LINKS
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FORMULA
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EXAMPLE
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For n = 3, the average of the gcd's is (gcd(1,3) + gcd(2,3) + gcd(3,3))/3 = (1 + 1 + 3)/3 = 5/3 and its numerator is a(3)=5.
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MATHEMATICA
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Table[Numerator[Sum[GCD[I, j], {j, 1, I}]/I], {I, 100}]
f[p_, e_] := e*(p - 1)/p + 1; a[n_] := Numerator[Times @@ f @@@ FactorInteger[n]]; Array[a, 100] (* Amiram Eldar, Apr 28 2023 *)
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PROG
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(Haskell) map numerator (map (\i -> sum (map (\j -> gcd i j) [1..i]) % i) [1..])
(PARI) a(n) = numerator(sum(i=1, n, gcd(i, n))/n); \\ Michel Marcus, Aug 08 2022
(PARI) a(n, f=factor(n))=my(k=prod(i=1, #f~, (f[i, 2]*(f[i, 1]-1)/f[i, 1] + 1)*f[i, 1]^f[i, 2])); k/gcd(k, n) \\ Charles R Greathouse IV, Sep 08 2022
(Python)
from math import prod, gcd
from sympy import factorint
f = factorint(n)
return (m:=prod((p-1)*e+p for p, e in f.items()))//gcd(prod(f), m) # Chai Wah Wu, Sep 08 2022
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CROSSREFS
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KEYWORD
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easy,frac,nonn
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AUTHOR
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STATUS
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approved
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