

A161942


Odd part of sum of divisors of n.


52



1, 3, 1, 7, 3, 3, 1, 15, 13, 9, 3, 7, 7, 3, 3, 31, 9, 39, 5, 21, 1, 9, 3, 15, 31, 21, 5, 7, 15, 9, 1, 63, 3, 27, 3, 91, 19, 15, 7, 45, 21, 3, 11, 21, 39, 9, 3, 31, 57, 93, 9, 49, 27, 15, 9, 15, 5, 45, 15, 21, 31, 3, 13, 127, 21, 9, 17, 63, 3, 9, 9, 195, 37, 57, 31, 35, 3, 21, 5, 93, 121, 63
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OFFSET

1,2


COMMENTS

It is conjectured that iteration of this function will always reach 1. This implies the nonexistence of odd perfect numbers. This is equivalent to the same question for A000593, which can be expressed as the sum of the divisors of the odd part of n.
Up to 20000000, there are only two odd numbers with a(n) and a(a(n)) both >= n: 81 and 18966025. See A162284.
For the nonexistence proof of odd perfect numbers, it is enough to show that this sequence has no fixed points beyond the initial one. This is equivalent to a similar condition given for A326042.  Antti Karttunen, Jun 17 2019


LINKS

Antti Karttunen, Table of n, a(n) for n = 1..16384
Index entries for sequences related to sigma(n)
Index entries for sequences related to sums of divisors


FORMULA

Multiplicative with a(p^e) = oddpart((p^{e+1}1)/(p1)), where oddpart(n) = A000265(n) is the largest odd divisor of n.
a(n) = A000265(A000203(n)).


MATHEMATICA

oddPart[n_] := n/2^IntegerExponent[n, 2]; a[n_] := oddPart[ DivisorSigma[1, n]]; Table[a[n], {n, 1, 82}] (* JeanFrançois Alcover, Sep 03 2012 *)


PROG

(PARI)
oddpart(n)=n/2^valuation(n, 2);
a(n)=oddpart(sigma(n));
(Scheme) (define (A161942 n) (A000265 (A000203 n))) ;; [For the implementations of A000203 and A000265, see under the respective entries].  Antti Karttunen, Nov 18 2017


CROSSREFS

Cf. A000265, A000203, A000593, A162284, A326042.
Sequence in context: A050227 A354095 A135540 * A247675 A053092 A212045
Adjacent sequences: A161939 A161940 A161941 * A161943 A161944 A161945


KEYWORD

easy,mult,nonn


AUTHOR

Franklin T. AdamsWatters, Jun 22 2009


STATUS

approved



