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A336701 Numbers k for which A000265(1+A000265(sigma(k))) is equal to A000265(1+k).

%I A336701 #56 Feb 16 2025 08:34:00
%S A336701 1,3,7,15,31,127,1023,8191,34335,57855,131071,524287,2147483647
%N A336701 Numbers k for which A000265(1+A000265(sigma(k))) is equal to A000265(1+k).
%C A336701 Numbers k such that A336698(k) [= A000265(1+A161942(k))] is equal to A000265(1+k).
%C A336701 Numbers k such that A337194(k) = 2^e * A000265(1+k), for some e >= 1, where that e = A337195(k).
%C A336701 Any odd perfect number would trivially satisfy this condition.
%C A336701 Also, all hypothetical quasiperfect numbers, numbers k that satisfy sigma(k) = 2k+1, would be members.
%C A336701 Question: Is A066175 a subsequence of this sequence?
%C A336701 From _Antti Karttunen_, Aug 23 2020: (Start)
%C A336701 Numbers k such that (1+k) = 2^e * A336698(k), for some e >= 0.
%C A336701 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).
%C A336701 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))).
%C A336701 (End)
%H A336701 Paolo Cattaneo, <a href="http://www.bdim.eu/item?fmt=pdf&amp;id=BUMI_1951_3_6_1_59_0">Sui numeri quasiperfetti</a>, Bollettino dell’Unione Matematica Italiana, Serie 3, Vol.6(1951), n.1, p. 59-62.
%H A336701 P. Hagis and G. L. Cohen, <a href="http://dx.doi.org/10.1017/S1446788700018401">Some Results Concerning Quasiperfect Numbers</a>, J. Austral. Math. Soc. Ser. A 33, 275-286, 1982.
%H A336701 V. Siva Rama Prasad and C. Sunitha, <a href="http://nntdm.net/papers/nntdm-23/NNTDM-23-3-073-078.pdf">On quasiperfect numbers</a>, Notes on Number Theory and Discrete Mathematics, Vol. 23, 2017, No. 3, 73-78.
%H A336701 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/QuasiperfectNumber.html">Quasiperfect Number</a>
%H A336701 <a href="/index/O#opnseqs">Index entries for sequences where odd perfect numbers must occur, if they exist at all</a>
%H A336701 <a href="/index/Si#SIGMAN">Index entries for sequences related to sigma(n)</a>
%t A336701 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 *)
%o A336701 (PARI)
%o A336701 A000265(n)  = (n>>valuation(n,2));
%o A336701 isA336701(n) = (A000265(1+A000265(sigma(n))) == A000265(1+n));
%Y A336701 Subsequence of A336700.
%Y A336701 Cf. A000203, A000265, A066175, A161942, A336698, A336699, A336937, A337194, A337195.
%Y A336701 Cf. A000668 (a subsequence).
%Y A336701 See also comments in A326042, A332223.
%K A336701 nonn,more
%O A336701 1,2
%A A336701 _Antti Karttunen_, Aug 02 2020