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A385138
The sum of divisors d of n such that n/d is a 5-rough number (A007310).
6
1, 2, 3, 4, 6, 6, 8, 8, 9, 12, 12, 12, 14, 16, 18, 16, 18, 18, 20, 24, 24, 24, 24, 24, 31, 28, 27, 32, 30, 36, 32, 32, 36, 36, 48, 36, 38, 40, 42, 48, 42, 48, 44, 48, 54, 48, 48, 48, 57, 62, 54, 56, 54, 54, 72, 64, 60, 60, 60, 72, 62, 64, 72, 64, 84, 72, 68, 72
OFFSET
1,2
LINKS
FORMULA
a(n) = A064987(n)/A385137(n).
Multiplicative with a(p^e) = p^e if p <= 3, and (p^(e+1)-1)/(p-1) if p >= 5.
In general, the sum of divisors d of n such that n/d is q-rough (not divisible by a prime smaller than q) is multiplicative with a(p^e) = p^e if p <= q, and (p^(e+1)-1)/(p-1) if p > q.
Dirichlet g.f.: zeta(s-1) * zeta(s) * ((1 - 1/2^s) * (1 - 1/3^s)).
In general, the sum of divisors d of n such that n/d is q-rough has Dirichlet g.f.: zeta(s-1) * zeta(s) * Product_{p prime <= q} (1 - 1/q^s).
Sum_{k=1..n} a(k) ~ (Pi^2/18)*n^2.
In general, the sum of divisors d of n such that n/d is prime(k)-rough has an average order c * n^2 / 2, where c = zeta(2) * A072045(k-1)/A072044(k-1) for k >= 2.
MATHEMATICA
f[p_, e_] := If[p > 3, (p^(e+1) - 1)/(p - 1), p^e]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]
PROG
(PARI) a(n) = {my(f = factor(n), p, e); prod(i = 1, #f~, p = f[i, 1]; e = f[i, 2]; if(p > 3, (p^(e + 1) - 1)/(p - 1), p^e)); }
CROSSREFS
The sum of divisors d of n such that n/d is: A001615 (squarefree), A002131 (odd), A069208 (powerful), A076752 (square), A129527 (power of 2), A254981 (cubefree), A244963 (nonsquarefree), A327626 (cube), A385134 (biquadratefree), A385135 (exponentially odd), A385136 (cubefull), A385137 (3-smooth), this sequence (5-rough), A385139 (exponentially 2^n).
Sequence in context: A214965 A389650 A350786 * A134361 A142727 A347388
KEYWORD
nonn,easy,mult
AUTHOR
Amiram Eldar, Jun 19 2025
STATUS
approved