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A324647
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Odd numbers k such that 2k is equal to A318468(k) (bitwise-AND of 2*k and sigma(k)).
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11
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1116225, 1245825, 1380825, 2127825, 10046025, 16813125, 203753025, 252880425, 408553425, 415433025, 740361825, 969523425, 1369580625, 1612924425, 1763305425, 2018027025, 2048985225, 2286684225, 3341556225, 3915517725, 3985769025, 4051698525, 7085469825, 7520472225
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OFFSET
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1,1
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COMMENTS
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If this sequence has no terms common with A324649 (A324897, A324898), or no terms common with A324727, then there are no odd perfect numbers.
First 22 terms factored:
1116225 = 3^2 * 5^2 * 11^2 * 41
1245825 = 3^2 * 5^2 * 7^2 * 113
1380825 = 3^2 * 5^2 * 19^2 * 17 [Here the unitary prime is not the largest]
2127825 = 3^2 * 5^2 * 7^2 * 193
10046025 = 3^4 * 5^2 * 11^2 * 41
16813125 = 3^2 * 5^4 * 7^2 * 61
203753025 = 3^2 * 5^2 * 7^2 * 18481
252880425 = 3^2 * 5^2 * 7^2 * 22937
408553425 = 3^2 * 5^2 * 7^2 * 37057
415433025 = 3^2 * 5^2 * 7^4 * 769
740361825 = 3^2 * 5^2 * 7^2 * 67153
969523425 = 3^4 * 5^2 * 13^2 * 2833
1369580625 = 3^2 * 5^4 * 7^2 * 4969
1612924425 = 3^2 * 5^2 * 7^2 * 146297
1763305425 = 3^2 * 5^2 * 7^2 * 159937
2018027025 = 3^2 * 5^2 * 7^2 * 183041
2048985225 = 3^2 * 5^2 * 7^2 * 185849
2286684225 = 3^2 * 5^2 * 7^2 * 207409
3341556225 = 3^2 * 5^2 * 7^2 * 303089
3915517725 = 3^4 * 5^2 * 7^2 * 39461
3985769025 = 3^4 * 5^2 * 7^2 * 40169
4051698525 = 3^2 * 5^2 * 7^2 * 367501.
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.
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
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LINKS
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PROG
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(PARI) for(n=1, oo, if((n%2)&&((2*n)==bitand(2*n, sigma(n))), print1(n, ", ")));
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CROSSREFS
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Cf. A191218, A228058, A318468, A324649, A324659, A324718, A324719, A324722, A324727, A324880, A324897, A324898, A325311.
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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