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 A161942 Odd part of sum of divisors of n. 27
 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 (list; graph; refs; listen; history; text; internal format)
 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 FORMULA Multiplicative with a(p^e) = oddpart((p^{e+1}-1)/(p-1)), 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}] (* Jean-François 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: A130330 A050227 A135540 * A247675 A053092 A212045 Adjacent sequences:  A161939 A161940 A161941 * A161943 A161944 A161945 KEYWORD easy,mult,nonn AUTHOR Franklin T. Adams-Watters, Jun 22 2009 STATUS approved

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Last modified April 16 17:01 EDT 2021. Contains 343050 sequences. (Running on oeis4.)