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 A336701 Numbers k for which A000265(1+A000265(sigma(k))) is equal to A000265(1+k). 5
 1, 3, 7, 15, 31, 127, 1023, 8191, 34335, 57855, 131071, 524287, 2147483647 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS Numbers k such that A336698(k) [= A000265(1+A161942(k))] is equal to A000265(1+k). Numbers k such that A337194(k) = 2^e * A000265(1+k), for some e >= 1, where that e = A337195(k). Any odd perfect number would trivially satisfy this condition. Also, all hypothetical quasiperfect numbers, numbers k that satisfy sigma(k) = 2k+1, would be members. Question: Is A066175 a subsequence of this sequence? From Antti Karttunen, Aug 23 2020: (Start) Numbers k such that (1+k) = 2^e * A336698(k), for some e >= 0. Thus numbers k such that for some e >= 0, (1+k) = 2^(e-A337195(k)) * A337194(k), or equally, that A337194(k) = 2^(A337195(k)-e) * (1+k). Conjecture: There are no even terms. This is equivalent to claim that there are no k such that A336698(k) = 1+k: If we assume that k is even, then in above equations we set e=0, and the requirement will then become that A337194(k) = 2^A337195(k)*(1+k), thus 1+k = A336698(k) = A000265(1+A000265(sigma(k))). (End) LINKS Table of n, a(n) for n=1..13. Paolo Cattaneo, Sui numeri quasiperfetti, Bollettino dell’Unione Matematica Italiana, Serie 3, Vol.6(1951), n.1, p. 59-62. P. Hagis and G. L. Cohen, Some Results Concerning Quasiperfect Numbers, J. Austral. Math. Soc. Ser. A 33, 275-286, 1982. V. Siva Rama Prasad and C. Sunitha, On quasiperfect numbers, Notes on Number Theory and Discrete Mathematics, Vol. 23, 2017, No. 3, 73-78. Eric Weisstein's World of Mathematics, Quasiperfect Number Index entries for sequences where odd perfect numbers must occur, if they exist at all Index entries for sequences related to sigma(n) MATHEMATICA Block[{f}, f[n_] := n/2^IntegerExponent[n, 2]; Select[Range[2^20], f[1 + f[DivisorSigma[1, #]]] == f[1 + #] &] ] (* Michael De Vlieger, Aug 22 2020 *) PROG (PARI) A000265(n) = (n>>valuation(n, 2)); isA336701(n) = (A000265(1+A000265(sigma(n))) == A000265(1+n)); CROSSREFS Subsequence of A336700. Cf. A000203, A000265, A066175, A161942, A336698, A336699, A336937, A337194, A337195. Cf. A000668 (a subsequence). See also comments in A326042, A332223. Sequence in context: A326518 A147285 A147250 * A066175 A132978 A336976 Adjacent sequences: A336698 A336699 A336700 * A336702 A336703 A336704 KEYWORD nonn,more AUTHOR Antti Karttunen, Aug 02 2020 STATUS approved

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Last modified May 27 22:51 EDT 2023. Contains 362992 sequences. (Running on oeis4.)