A324898 Odd numbers k such that sigma(k) is congruent to 2 modulo 4 and k = A318458(k), where A318458(k) is bitwise-AND of k and sigma(k)-k.

236925, 3847725, 51122925, 69468525, 151141725, 154669725, 269748225, 344211525, 415565325, 445817925, 551569725, 1111904325, 1112565825, 1113756525, 1175717025, 1400045625, 1631666925, 1695170925, 1820873925, 1915847325, 1946981925, 2179080225, 2321121825, 2453690925, 2460041325, 2491740225, 3223500525, 3493517445, 3775103325

Offset: 1

Author

Antti Karttunen, Apr 19 2019

Keywords

• nonn

Comments

If this sequence has no common terms with A324647, or no terms common with A324727, then there are no odd perfect numbers.

The first 29 terms factored:

236925 = 3^6 * 5^2 * 13,

3847725 = 3^2 * 5^2 * 7^2 * 349,

51122925 = 3^2 * 5^2 * 7^2 * 4637,

69468525 = 3^2 * 5^2 * 7^2 * 6301,

151141725 = 3^2 * 5^2 * 7^2 * 13709,

154669725 = 3^2 * 5^2 * 7^2 * 14029,

269748225 = 3^6 * 5^2 * 19^2 * 41,

344211525 = 3^4 * 5^2 * 7^2 * 3469,

415565325 = 3^2 * 5^2 * 7^2 * 37693,

445817925 = 3^4 * 5^2 * 7^2 * 4493,

551569725 = 3^2 * 5^2 * 7^4 * 1021,

1111904325 = 3^2 * 5^2 * 7^2 * 100853,

1112565825 = 3^2 * 5^2 * 7^2 * 100913,

1113756525 = 3^2 * 5^2 * 7^2 * 101021,

1175717025 = 3^4 * 5^2 * 7^2 * 17^2 * 41,

1400045625 = 3^2 * 5^4 * 11^4 * 17,

1631666925 = 3^2 * 5^2 * 7^2 * 147997,

1695170925 = 3^2 * 5^2 * 7^2 * 153757,

1820873925 = 3^4 * 5^2 * 13 * 263^2, [Here the unitary prime is not the largest]

1915847325 = 3^2 * 5^2 * 7^2 * 173773,

1946981925 = 3^2 * 5^2 * 7^2 * 176597,

2179080225 = 3^4 * 5^2 * 7^2 * 21961,

2321121825 = 3^4 * 5^2 * 11^2 * 9473,

2453690925 = 3^2 * 5^2 * 7^2 * 222557,

2460041325 = 3^2 * 5^2 * 7^2 * 223133,

2491740225 = 3^6 * 5^2 * 13^2 * 809,

3223500525 = 3^2 * 5^2 * 7^2 * 292381,

3493517445 = 3^6 * 5^1 * 11^2 * 89^2, [Here the unitary prime is not the largest]

3775103325 = 3^2 * 5^2 * 7^2 * 342413.

Subsequence of A228058 provided this sequence does not contain any prime powers. - Antti Karttunen, Jun 17 2019

Sequence contains no prime powers up to 10^20. I believe any prime powers must be of the form (4k+1)^(4e+1), in which case I have verified this up to 10^50. - Charles R Greathouse IV, Dec 08 2021

Links

• Index entries for sequences related to binary expansion of n
• Index entries for sequences where any odd perfect numbers must occur
• Index entries for sequences related to sigma(n)

Crossrefs

Intersection of A191218 and A324897, also intersection of A191218 and A324649.

Cf. A228058, A318458, A324647, A324727.

Programs

Mathematica

Select[Range[10^5, 10^8, 2], And[Mod[#2, 4] == 2, BitAnd[#1, #2 - #1] == #1] & @@ {#, DivisorSigma[1, #]} &] (* Michael De Vlieger, Jun 22 2019 *)

PARI

for(n=1, oo, if((n%2)&&2==((t=sigma(n))%4)&&(bitand(n, t-n)==n), print1(n,", ")));