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A336700 Numbers k such that the odd part of (1+k) divides (1 + odd part of sigma(k)).

%I A336700 #42 Feb 16 2025 08:34:00
%S A336700 1,3,7,15,31,63,127,255,511,1023,2047,2431,2943,3775,4095,8191,13311,
%T A336700 14335,16383,17407,21951,22527,32767,34335,44031,57855,65535,85375,
%U A336700 131071,204799,262143,376831,524287,923647,1048575,1562623,1632255,2056191,2097151,2744319,4194303,6815743,8388607,8781823,10059775,16777215
%N A336700 Numbers k such that the odd part of (1+k) divides (1 + odd part of sigma(k)).
%C A336700 Numbers k for which A337194(k) = 1+A161942(k) is a multiple of A000265(1+k).
%C A336700 Conjecture: After 1, all terms are of the form 4u+3 (in A004767). If this could be proved, then it would refute at once the existence of both the odd perfect numbers and the quasiperfect numbers. Concentrating on the latter is probably easier, as it is known that all quasiperfect numbers must be odd squares, thus k is of the form 4u+1, in which case the condition given in A336701 that A000265(1+A000265(sigma(k))) must be equal to A000265(1+k) reduces to a simpler form, A000265(1+sigma(k)) = (1+k)/2, and as k = s^2, with s odd, so (s^2 + 1)/2 should divide 1+sigma(s^2). Does that condition allow any other solutions than s=1 ? See A337339.
%H A336700 Antti Karttunen, <a href="/A336700/b336700.txt">Table of n, a(n) for n = 1..77; all terms < 2^32</a>
%H A336700 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 A336700 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 A336700 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 A336700 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/QuasiperfectNumber.html">Quasiperfect Number</a>
%H A336700 <a href="/index/O#opnseqs">Index entries for sequences where odd perfect numbers must occur, if they exist at all</a>
%H A336700 <a href="/index/Si#SIGMAN">Index entries for sequences related to sigma(n)</a>
%t A336700 Block[{f}, f[n_] := n/2^IntegerExponent[n, 2]; Select[Range[2^20], Mod[f[1 + f[DivisorSigma[1, #]]], f[1 + #]] == 0 &] ] (* _Michael De Vlieger_, Aug 22 2020 *)
%o A336700 (PARI)
%o A336700 A000265(n) = (n>>valuation(n,2));
%o A336700 isA336700(n) = !((1+A000265(sigma(n)))%A000265(1+n));
%Y A336700 Cf. A000203, A000265, A004767, A016754, A161942, A228058, A336697, A336698, A337339, A337342.
%Y A336700 Subsequences: A000225, A336701 (terms where the quotient is a power of 2).
%K A336700 nonn
%O A336700 1,2
%A A336700 _Antti Karttunen_, Aug 02 2020