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 A324647 Odd numbers k such that 2k is equal to A318468(k) (bitwise-AND of 2*k and sigma(k)). 11

%I

%S 1116225,1245825,1380825,2127825,10046025,16813125,203753025,

%T 252880425,408553425,415433025,740361825,969523425,1369580625,

%U 1612924425,1763305425,2018027025,2048985225,2286684225,3341556225,3915517725,3985769025,4051698525,7085469825,7520472225

%N Odd numbers k such that 2k is equal to A318468(k) (bitwise-AND of 2*k and sigma(k)).

%C 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 First 22 terms factored:

%C 1116225 = 3^2 * 5^2 * 11^2 * 41

%C 1245825 = 3^2 * 5^2 * 7^2 * 113

%C 1380825 = 3^2 * 5^2 * 19^2 * 17 [Here the unitary prime is not the largest]

%C 2127825 = 3^2 * 5^2 * 7^2 * 193

%C 10046025 = 3^4 * 5^2 * 11^2 * 41

%C 16813125 = 3^2 * 5^4 * 7^2 * 61

%C 203753025 = 3^2 * 5^2 * 7^2 * 18481

%C 252880425 = 3^2 * 5^2 * 7^2 * 22937

%C 408553425 = 3^2 * 5^2 * 7^2 * 37057

%C 415433025 = 3^2 * 5^2 * 7^4 * 769

%C 740361825 = 3^2 * 5^2 * 7^2 * 67153

%C 969523425 = 3^4 * 5^2 * 13^2 * 2833

%C 1369580625 = 3^2 * 5^4 * 7^2 * 4969

%C 1612924425 = 3^2 * 5^2 * 7^2 * 146297

%C 1763305425 = 3^2 * 5^2 * 7^2 * 159937

%C 2018027025 = 3^2 * 5^2 * 7^2 * 183041

%C 2048985225 = 3^2 * 5^2 * 7^2 * 185849

%C 2286684225 = 3^2 * 5^2 * 7^2 * 207409

%C 3341556225 = 3^2 * 5^2 * 7^2 * 303089

%C 3915517725 = 3^4 * 5^2 * 7^2 * 39461

%C 3985769025 = 3^4 * 5^2 * 7^2 * 40169

%C 4051698525 = 3^2 * 5^2 * 7^2 * 367501.

%C 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 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

%H Giovanni Resta, <a href="/A324647/b324647.txt">Table of n, a(n) for n = 1..500</a>

%H Charles Greathouse and Eric W. Weisstein, <a href="http://mathworld.wolfram.com/OddPerfectNumber.html">MathWorld: Odd perfect number</a>

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Perfect_number#Odd_perfect_numbers">Perfect number: Odd perfect numbers</a>

%H <a href="/index/O#opnseqs">Index entries for sequences where any odd perfect numbers must occur</a>

%o (PARI) for(n=1,oo,if((n%2)&&((2*n)==bitand(2*n,sigma(n))),print1(n,", ")));

%Y Odd terms of A324652.

%Y Cf. A191218, A228058, A318468, A324649, A324659, A324718, A324719, A324722, A324727, A324880, A324897, A324898, A325311.

%K nonn

%O 1,1

%A _Antti Karttunen_, Mar 14 2019

%E a(23)-a(24) from _Giovanni Resta_, Apr 21 2019

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Last modified January 20 11:11 EST 2020. Contains 331083 sequences. (Running on oeis4.)