A324649 -id:A324649 - cp's OEIS frontend

A324647 Odd numbers k such that 2k is equal to bitwise-AND of 2k 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

Antti Karttunen, Mar 14 2019

nonn

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)

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

Odd terms of A324652.

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

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

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

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

A324652 Numbers k such that A318468(k) (bitwise-AND of 2k and sigma(k)) is equal to 2k.

Original entry on oeis.org

6, 12, 18, 20, 24, 28, 36, 40, 48, 56, 80, 88, 96, 100, 104, 112, 160, 176, 192, 196, 200, 204, 208, 220, 224, 260, 264, 272, 304, 320, 336, 352, 368, 384, 392, 416, 448, 464, 496, 544, 550, 580, 608, 640, 648, 650, 672, 704, 736, 768, 784, 832, 896, 928, 992, 1032, 1040, 1044, 1056, 1060, 1068, 1088, 1104, 1120, 1184, 1216

Offset: 1

Antti Karttunen, Mar 14 2019

nonn

Positions of zeros in A324658, fixed points of A324659.
Intersection with A324649 gives A324643.
Intersection with A324726 gives A000396.
In the range 1..2^32 there are only 22 odd terms. See A324647.

Cf. A000203, A191218, A318468, A324647, A324649, A324658, A324659, A324722, A324726.

Some subsequences: A000396, A324643, A324647 (the odd terms).

Select[Range[2000], 2*# == BitAnd[2*#, DivisorSigma[1, #]] &] (* Paolo Xausa, Mar 11 2024 *)

for(n=1,oo,if((n+n)==bitand(2*n,sigma(n)), print1(n, ", ")))

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.

Original entry on oeis.org

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

Antti Karttunen, Apr 19 2019

nonn

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

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

Cf. A228058, A318458, A324647, A324727.

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 *)

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

A324643 Numbers k such that bitand(2k,sigma(k))/2 = k = bitand(k,sigma(k)-k), where bitand is bitwise-AND, A004198.

Original entry on oeis.org

6, 20, 28, 36, 88, 100, 104, 264, 272, 304, 368, 392, 464, 496, 550, 784, 1032, 1040, 1044, 1056, 1068, 1104, 1120, 1184, 1232, 1312, 1376, 1504, 1696, 1888, 1952, 2140, 3222, 4100, 4128, 4160, 4288, 4512, 4544, 4624, 4640, 4672, 5056, 5312, 5696, 6208, 6328, 6464, 6592, 6808, 6848, 6976, 7232, 7304, 8128, 8288, 8968, 9256, 10184

Offset: 1

Antti Karttunen, Mar 14 2019

Numbers k for which k = A318458(k)/2 = A318468(k).
Intersection of A324649 and A324652.
It is conjectured that there are no odd terms in this sequence, which is equivalent to the conjecture that there are no odd perfect numbers.
Question: Where do the densest clusters of terms occur? See also the scatter plot. - Antti Karttunen, Mar 12 2024
As A324649 and A324652 are both subsequences of nondeficient numbers (A023196), also this sequence is, which stems from the "monotonic property" of bitwise-and. - Antti Karttunen, Jan 08 2025

Antti Karttunen, Table of n, a(n) for n = 1..17020
Michael De Vlieger, Log log scatterplot of a(n), n = 1..17020, labeling cusps in the plot.
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)

Cf. A004198, A318458, A318468.
Intersection of A324649 and A324652.
Subsequence of A023196 and of A324639.

Select[Range[10^4], Block[{s = DivisorSigma[1, #]}, # == BitAnd[#, s-#] && 2*# == BitAnd[2*#, s]] &] (* Paolo Xausa, Mar 11 2024 *)

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

A324648 a(n) = n - A318458(n), where A318458(n) is bitwise-AND of n and the sum of proper divisors of n (sigma(n)-n).

Original entry on oeis.org

1, 2, 2, 4, 4, 0, 6, 8, 9, 2, 10, 12, 12, 4, 6, 16, 16, 2, 18, 0, 20, 16, 22, 24, 25, 10, 18, 0, 28, 20, 30, 32, 32, 34, 34, 0, 36, 32, 38, 8, 40, 8, 42, 4, 12, 36, 46, 48, 49, 16, 34, 16, 52, 52, 38, 56, 40, 26, 58, 16, 60, 28, 22, 64, 64, 0, 66, 68, 68, 4, 70, 0, 72, 66, 74, 12, 76, 4, 78, 16, 81, 82, 82, 80, 64, 80, 86, 0, 88

Offset: 1

Antti Karttunen, Mar 14 2019

Antti Karttunen, Table of n, a(n) for n = 1..16384
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537
Index entries for sequences related to binary expansion of n
Index entries for sequences related to sigma(n)

Cf. A001065, A004198, A318458, A324658, A324649 (positions of zeros).

Array[# - BitAnd[#, DivisorSigma[1, #] - #] &, 100] (* Paolo Xausa, Mar 12 2024 *)

A318458(n) = bitand(n,sigma(n)-n);
A324648(n) = (n-A318458(n));

PARI
```
A324648(n) = (n-bitand(n,sigma(n)-n));
```

a(n) = n - A318458(n).

A324897 Odd numbers k such that A318458(k) (bitwise-AND of k and sigma(k)-k) is equal to k.

Original entry on oeis.org

7425, 76545, 92565, 236925, 831105, 954765, 1401345, 2011905, 2048445, 2129985, 2253825, 2445345, 2621745, 2974725, 3283245, 3847725, 5709825, 6447105, 8422785, 8503425, 8945685, 10781505, 12488385, 13470345, 14322945, 15213825, 15340545, 19470465, 19502145, 20075265, 22749825, 25740225, 25756605, 26215245, 27009045

Offset: 1

Antti Karttunen, Apr 19 2019

nonn

If this sequence has no common terms with A324647, or no terms common with A324727, then there are no odd perfect numbers.
The first 16 terms factored:
      7425 = 3^3 * 5^2 * 11,
     76545 = 3^7 * 5 * 7,
     92565 = 3^2 * 5 * 11^2 * 17,
    236925 = 3^6 * 5^2 * 13,
    831105 = 3^2 * 5 * 11 * 23 * 73,
    954765 = 3^2 * 5 * 7^2 * 433,
   1401345 = 3^2 * 5 * 11 * 19 * 149,
   2011905 = 3^3 * 5 * 7 * 2129,
   2048445 = 3^2 * 5 * 7^2 * 929,
   2129985 = 3^2 * 5 * 11 * 13 * 331,
   2253825 = 3^5 * 5^2 * 7 * 53,
   2445345 = 3^2 * 5 * 7^2 * 1109,
   2621745 = 3^2 * 5 * 7^2 * 29 * 41,
   2974725 = 3^4 * 5^2 * 13 * 113,
   3283245 = 3^2 * 5 * 7^2 * 1489,
   3847725 = 3^2 * 5^2 * 7^2 * 349.

Subsequence of A324649.

Cf. A318458, A324647, A324898 (a subsequence).

Select[Range[1, 10^7, 2], BitAnd[#, DivisorSigma[1, #] - #] == # &] (* Michael De Vlieger, Jun 22 2019, after Vincenzo Librandi at A318458 *)

isok(k) = (k%2) && (bitand(k, sigma(k)-k) == k); \\ Michel Marcus, Jul 18 2021

cp's OEIS Frontend

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

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A324652 Numbers k such that A318468(k) (bitwise-AND of 2*k and sigma(k)) is equal to 2*k.

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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.

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A324643 Numbers k such that bitand(2k,sigma(k))/2 = k = bitand(k,sigma(k)-k), where bitand is bitwise-AND, A004198.

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A324648 a(n) = n - A318458(n), where A318458(n) is bitwise-AND of n and the sum of proper divisors of n (sigma(n)-n).

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A324897 Odd numbers k such that A318458(k) (bitwise-AND of k and sigma(k)-k) is equal to k.

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

A324652 Numbers k such that A318468(k) (bitwise-AND of 2k and sigma(k)) is equal to 2k.