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A322965
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Numerator of Sum_{d | n} 1/rad(d).
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3
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1, 3, 4, 2, 6, 2, 8, 5, 5, 9, 12, 8, 14, 12, 8, 3, 18, 5, 20, 12, 32, 18, 24, 10, 7, 21, 2, 16, 30, 12, 32, 7, 16, 27, 48, 10, 38, 30, 56, 3, 42, 16, 44, 24, 2, 36, 48, 4, 9, 21, 24, 28, 54, 3, 72, 20, 80, 45, 60, 16, 62, 48, 40, 4, 84, 24, 68, 36, 32, 72, 72, 25, 74, 57, 28
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OFFSET
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1,2
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COMMENTS
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Let rad(n) be the radical of n, which equals the product of all prime factors of n (A007947). Let g(n) = 1/rad(n) and let f(n) = Sum_{d | n} g(d). This is a multiplicative function whose value on a prime power is f(p^k) = 1 + k/p. Hence f is a weighted divisor-counting function that weights divisors d higher when they have few and small prime divisors themselves. The sequence a(n) lists the numerators of the fractions f(n) in lowest terms.
If p is prime, then a(p^k) = p+k if p does not divide k, 1 + k/p if it does. In particular, a(p^p) = 2. - Robert Israel, Jan 25 2019
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LINKS
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FORMULA
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Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k)/A322966(k) = zeta(2)*zeta(3)/zeta(6) (A082695). - Amiram Eldar, Dec 09 2023
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EXAMPLE
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The divisors of 12 are 1,2,3,4,6,12, so f(12) = 1 + (1/2) + (1/3) + (1/2) + (1/6) + (1/6) = 8/3 and a(12) = 8. Alternately, since f is multiplicative, f(12) = f(4)*f(3) = (1+2/2)*(1+1/3) = 8/3.
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MAPLE
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rad:= n -> convert(numtheory:-factorset(n), `*`):
f:= proc(n) numer(add(1/rad(d), d=numtheory:-divisors(n))) end proc:
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MATHEMATICA
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Array[Numerator@ DivisorSum[#, 1/Apply[Times, FactorInteger[#][[All, 1]]] &] &, 71] (* Michael De Vlieger, Jan 19 2019 *)
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PROG
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(PARI) rad(n) = factorback(factor(n)[, 1]); \\ A007947
a(n) = numerator(sumdiv(n, d, 1/rad(d))); \\ Michel Marcus, Jan 10 2019
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CROSSREFS
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Numbers n where f(n) increases to a record: A322447.
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KEYWORD
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frac,nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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