login
A336702
Numbers whose abundancy index is a power of 2.
37
1, 6, 28, 496, 8128, 30240, 32760, 2178540, 23569920, 33550336, 45532800, 142990848, 1379454720, 8589869056, 43861478400, 66433720320, 137438691328, 153003540480, 403031236608, 704575228896, 181742883469056, 6088728021160320, 14942123276641920, 20158185857531904, 275502900594021408, 622286506811515392, 2305843008139952128
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)
Odd terms of this sequence are given by the intersection of A349169 and A349174.
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)
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
Cf. A000396, A027687 (subsequences).
Subsequence of A007691, and after 1, also subsequence of A325637.
Union with {2} gives the positions of zeros in A347381.
Sequence in context: A370789 A351440 A325637 * A325654 A201186 A060286
KEYWORD
nonn
AUTHOR
Antti Karttunen, Aug 05 2020
STATUS
approved