A351569 - OEIS

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A351569

Sum of divisors of the largest unitary divisor of n that is an exponentially odd number.

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

LINKS

Antti Karttunen, Table of n, a(n) for n = 1..20000

Index entries for sequences related to sigma(n).

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

Adjacent sequences: A351566 A351567 A351568 * A351570 A351571 A351572

KEYWORD

nonn,mult

AUTHOR

Antti Karttunen, Feb 23 2022

STATUS

approved