OFFSET

1,2

COMMENTS

Apart from missing 2, this sequence gives all numbers k such that the binary expansion of A156552(k) is a prefix of that of A156552(sigma(k)), that is, for k > 1, numbers k for which sigma(k) is a descendant of k in A005940-tree. This follows because of the two transitions x -> A005843(x) (doubling) and x -> A003961(x) (prime shift) used to generate descendants in A005940-tree, using A003961 at any step of the process will ruin the chances of encountering sigma(k) anywhere further down that subtree.

Proof: Any left child in A005940 (i.e., A003961(k) for k) is larger than sigma(k), for any k > 2 [see A286385 for a proof], and A003961(n) > n for all n > 1. Thus, apart from A003961(2) = 3 = sigma(2), A003961^t(k) > sigma(k), where A003961^t means t-fold application of prime shift, here with t >= 1. On the other hand, sigma(2n) > sigma(n) for all n, thus taking first some doubling steps before a run of one or more prime shift steps will not rescue us, as neither will taking further doubling steps after a bout of prime shifts.

The first terms of A325637 not included in this sequence are 154345556085770649600 and 9186050031556349952000, as they have abundancy index 6.

From Antti Karttunen, Nov 29 2021: (Start)

A064989 applied to the odd terms of this sequence gives the fixed points of A326042, i.e., the positions of zeros in A348736, and a subset of the positions of ones in A348941.

Odd terms of this sequence form a subsequence of A348943, but should occur neither in A348748 nor in A348749.

(End)

LINKS

Antti Karttunen, Table of n, a(n) for n = 1..769 (computed from the b-file of A007691 prepared by T. D. Noe, using Flammenkamp's data)

EXAMPLE

For 30240, sigma(30240) = 120960 = 4*30240, therefore, as sigma(k)/k = 2^2, a power of two, 30240 is present.

PROG

(PARI) isA336702(n) = { my(r=sigma(n)/n); (1==denominator(r)&&!bitand(r, r-1)); }; \\ (Corrected) - Antti Karttunen, Aug 31 2021

CROSSREFS

KEYWORD

nonn

AUTHOR

Antti Karttunen, Aug 05 2020

STATUS

approved