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

1116225, 1245825, 1380825, 2127825, 10046025, 16813125, 203753025, 252880425, 408553425, 415433025, 740361825, 969523425, 1369580625, 1612924425, 1763305425, 2018027025, 2048985225, 2286684225, 3341556225, 3915517725, 3985769025, 4051698525, 7085469825, 7520472225

Offset: 1

Author

Antti Karttunen, Mar 14 2019

Keywords

• nonn

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

From Antti Karttunen, Jan 13 2025: (Start)

Because of the "monotonic property" of bitwise-and, this is a subsequence of nondeficient numbers (A023196).

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.

(End)

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

Crossrefs

Odd terms of A324652.

Cf. A191218, A228058, A318468, A324649, A324659, A324718, A324719, A324722, A324727, A324880, A324897, A324898, A325311, A379490 (square terms).

Programs

PARI

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

Formula

{Odd k such that 2k = A318468(k)}.

Extensions

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