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A351569
Sum of divisors of the largest unitary divisor of n that is an exponentially odd number.
8
1, 3, 4, 1, 6, 12, 8, 15, 1, 18, 12, 4, 14, 24, 24, 1, 18, 3, 20, 6, 32, 36, 24, 60, 1, 42, 40, 8, 30, 72, 32, 63, 48, 54, 48, 1, 38, 60, 56, 90, 42, 96, 44, 12, 6, 72, 48, 4, 1, 3, 72, 14, 54, 120, 72, 120, 80, 90, 60, 24, 62, 96, 8, 1, 84, 144, 68, 18, 96, 144, 72, 15, 74, 114, 4, 20, 96, 168, 80, 6, 1, 126, 84
OFFSET
1,2
FORMULA
Multiplicative with a(p^e) = (p^(e+1)-1)/(p-1) if e is odd and 1 otherwise.
a(n) = A000203(A350389(n)).
a(n) = A000203(n) / A351568(n).
Sum_{k=1..n} a(k) ~ c * n^2, where c = zeta(4)/2 = Pi^4/180 = 0.541161... . - Amiram Eldar, Nov 20 2022
Dirichlet g.f.: zeta(2*s) * zeta(2*s-2) * Product_{p prime} (1 + 1/p^(s-1) + 1/p^s - 1/p^(2*s-2)). - Amiram Eldar, Sep 03 2023
MATHEMATICA
f[p_, e_] := If[OddQ[e], (p^(e + 1) - 1)/(p - 1), 1]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Feb 23 2022 *)
PROG
(PARI)
A350389(n) = { my(m=1, f=factor(n)); for(k=1, #f~, if(1==(f[k, 2]%2), m *= (f[k, 1]^f[k, 2]))); (m); };
A351569(n) = sigma(A350389(n));
(Python)
from math import prod
from sympy import factorint
def A351569(n): return prod((p**(e+1)-1)//(p-1) if e % 2 else 1 for p, e in factorint(n).items()) # Chai Wah Wu, Feb 24 2022
CROSSREFS
Cf. A000203, A013662, A028982 (positions of odd terms), A268335 (exponentially odd numbers), A350389, A351568, A351571.
Coincides with A001615 on squarefree numbers, A005117.
Sequence in context: A358346 A348733 A368471 * A163762 A347084 A226776
KEYWORD
nonn,mult
AUTHOR
Antti Karttunen, Feb 23 2022
STATUS
approved