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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
1116225, 1245825, 1380825, 2127825, 10046025, 16813125, 203753025, 252880425, 408553425, 415433025, 740361825, 969523425, 1369580625, 1612924425, 1763305425, 2018027025, 2048985225, 2286684225, 3341556225, 3915517725, 3985769025, 4051698525, 7085469825, 7520472225 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

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

LINKS

Giovanni Resta, Table of n, a(n) for n = 1..500

Charles Greathouse and Eric W. Weisstein, MathWorld: Odd perfect number

Wikipedia, Perfect number: Odd perfect numbers

Index entries for sequences where any odd perfect numbers must occur

PROG

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

CROSSREFS

Odd terms of A324652.

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

Sequence in context: A205916 A205908 A258679 * A245662 A186066 A059121

Adjacent sequences:  A324644 A324645 A324646 * A324648 A324649 A324650

KEYWORD

nonn

AUTHOR

Antti Karttunen, Mar 14 2019

EXTENSIONS

a(23)-a(24) from Giovanni Resta, Apr 21 2019

STATUS

approved

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Last modified December 14 22:42 EST 2019. Contains 329987 sequences. (Running on oeis4.)