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A324647 Odd numbers k such that 2*k is equal to bitwise-AND of 2*k and sigma(k).

%I A324647 #75 Feb 16 2025 08:33:57
%S A324647 1116225,1245825,1380825,2127825,10046025,16813125,203753025,
%T A324647 252880425,408553425,415433025,740361825,969523425,1369580625,
%U A324647 1612924425,1763305425,2018027025,2048985225,2286684225,3341556225,3915517725,3985769025,4051698525,7085469825,7520472225
%N A324647 Odd numbers k such that 2*k is equal to bitwise-AND of 2*k and sigma(k).
%C A324647 If this sequence has no terms common with A324649 (A324897, A324898), or no terms common with A324727, then there are no odd perfect numbers.
%C A324647 First 22 terms factored:
%C A324647      1116225 = 3^2 * 5^2 * 11^2 * 41
%C A324647      1245825 = 3^2 * 5^2 *  7^2 * 113
%C A324647      1380825 = 3^2 * 5^2 * 19^2 * 17    [Here the unitary prime is not the largest]
%C A324647      2127825 = 3^2 * 5^2 *  7^2 * 193
%C A324647     10046025 = 3^4 * 5^2 * 11^2 * 41
%C A324647     16813125 = 3^2 * 5^4 *  7^2 * 61
%C A324647    203753025 = 3^2 * 5^2 *  7^2 * 18481
%C A324647    252880425 = 3^2 * 5^2 *  7^2 * 22937
%C A324647    408553425 = 3^2 * 5^2 *  7^2 * 37057
%C A324647    415433025 = 3^2 * 5^2 *  7^4 * 769
%C A324647    740361825 = 3^2 * 5^2 *  7^2 * 67153
%C A324647    969523425 = 3^4 * 5^2 * 13^2 * 2833
%C A324647   1369580625 = 3^2 * 5^4 *  7^2 * 4969
%C A324647   1612924425 = 3^2 * 5^2 *  7^2 * 146297
%C A324647   1763305425 = 3^2 * 5^2 *  7^2 * 159937
%C A324647   2018027025 = 3^2 * 5^2 *  7^2 * 183041
%C A324647   2048985225 = 3^2 * 5^2 *  7^2 * 185849
%C A324647   2286684225 = 3^2 * 5^2 *  7^2 * 207409
%C A324647   3341556225 = 3^2 * 5^2 *  7^2 * 303089
%C A324647   3915517725 = 3^4 * 5^2 *  7^2 * 39461
%C A324647   3985769025 = 3^4 * 5^2 *  7^2 * 40169
%C A324647   4051698525 = 3^2 * 5^2 *  7^2 * 367501.
%C A324647 Compare the above factorizations to the various constraints listed for odd perfect numbers in the Wikipedia article. However, this is NOT a subsequence of A191218 (A228058), see below.
%C A324647 The first terms that do not belong to A191218 are 399736269009 = (3 * 7^2 * 11 * 17 * 23)^2 and 1013616036225 = (3^2 * 5 * 13 * 1721)^2, that both occur instead in A325311. The first terms with omega(n) <> 4 are 9315603297, 60452246925, 68923392525, and 112206463425. They factor as 3^2 * 7^2 * 11^2 * 13^2 * 1033, 3^2 * 5^2 * 7^2 * 17^2 * 18973, 3^2 * 5^2 * 13^2 * 19^2 * 5021, 3^2 * 5^2 * 7^2 * 199^2 * 257. - _Giovanni Resta_, Apr 21 2019
%C A324647 From _Antti Karttunen_, Jan 13 2025: (Start)
%C A324647 Because of the "monotonic property" of bitwise-and, this is a subsequence of nondeficient numbers (A023196).
%C A324647 Both odd perfect numbers, and quasiperfect numbers, if such numbers exist at all, would satisfy the condition for being included in this sequence. Furthermore, any term must be either an odd square with an odd abundancy (in A156942), which subset is given in A379490 (where quasiperfect numbers must thus reside, if they exist), or be included in A228058, i.e., satisfy the Euler's criteria for odd perfect numbers.
%C A324647 (End)
%H A324647 Giovanni Resta, <a href="/A324647/b324647.txt">Table of n, a(n) for n = 1..500</a>
%H A324647 Charles Greathouse and Eric W. Weisstein, <a href="https://mathworld.wolfram.com/OddPerfectNumber.html">MathWorld: Odd perfect number</a>
%H A324647 Wikipedia, <a href="https://en.wikipedia.org/wiki/Perfect_number#Odd_perfect_numbers">Perfect number: Odd perfect numbers</a>
%H A324647 <a href="/index/O#opnseqs">Index entries for sequences where any odd perfect numbers must occur</a>
%F A324647 {Odd k such that 2k = A318468(k)}.
%o A324647 (PARI) for(n=1,oo,if((n%2)&&((2*n)==bitand(2*n,sigma(n))),print1(n,", ")));
%Y A324647 Odd terms of A324652.
%Y A324647 Cf. A191218, A228058, A318468, A324649, A324659, A324718, A324719, A324722, A324727, A324880, A324897, A324898, A325311, A379490 (square terms).
%K A324647 nonn
%O A324647 1,1
%A A324647 _Antti Karttunen_, Mar 14 2019
%E A324647 a(23)-a(24) from _Giovanni Resta_, Apr 21 2019