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A365209
The sum of divisors d of n such that gcd(d, n/d) is a 3-smooth number (A003586).
2
1, 3, 4, 7, 6, 12, 8, 15, 13, 18, 12, 28, 14, 24, 24, 31, 18, 39, 20, 42, 32, 36, 24, 60, 26, 42, 40, 56, 30, 72, 32, 63, 48, 54, 48, 91, 38, 60, 56, 90, 42, 96, 44, 84, 78, 72, 48, 124, 50, 78, 72, 98, 54, 120, 72, 120, 80, 90, 60, 168, 62, 96, 104, 127, 84, 144
OFFSET
1,2
COMMENTS
First differs from A000005 at n = 25.
The number of these divisors is A365208(n).
LINKS
FORMULA
Multiplicative with a(p^e) = (p^(e+1)-1)/(p-1) for p = 2 or 3, and a(p^e) = 1 + p^e for a prime p >= 5.
a(n) <= A000203(n), with equality if and only if n is not divisible by a square of a prime >= 5.
a(n) >= A034448(n), with equality if and only if n is neither divisible by 4 nor by 9.
a(n) = A000203(A065331(n)) * A034448(A065330(n)).
Dirichlet g.f.: (4^s/(4^s-2)) * (9^s/(9^s-3)) * zeta(s)*zeta(s-1)/zeta(2*s-1).
Sum_{k=1..n} a(k) ~ c * n^2, where c = (54/91) * zeta(2)/zeta(3) = (54/91) * A306633 = 0.812037... .
MATHEMATICA
f[p_, e_] := If[p <= 3, (p^(e+1)-1)/(p-1), 1 + p^e]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]
PROG
(PARI) a(n) = {my(f = factor(n)); prod(i = 1, #f~, if(f[i, 1] <= 3, (f[i, 1]^(f[i, 2]+1)-1)/(f[i, 1]-1), 1 + f[i, 1]^f[i, 2])); }
KEYWORD
nonn,easy,mult
AUTHOR
Amiram Eldar, Aug 26 2023
STATUS
approved