A324647 -id:A324647 - cp's OEIS frontend

A228058 Odd numbers of the form p^(1+4k) * r^2, where p is prime of the form 1+4m, r > 1, and gcd(p,r) = 1. (Euler's criteria for odd perfect numbers).

45, 117, 153, 245, 261, 325, 333, 369, 405, 425, 477, 549, 605, 637, 657, 725, 801, 833, 845, 873, 909, 925, 981, 1017, 1025, 1053, 1233, 1325, 1341, 1377, 1413, 1421, 1445, 1525, 1557, 1573, 1629, 1737, 1773, 1805, 1813, 1825, 2009, 2057, 2061, 2097, 2169

Offset: 1

T. D. Noe, Aug 13 2013

It has been proved that if an odd perfect number exists, it belongs to this sequence. The first term of the form p^5 * n^2 is 28125 = 5^5 * 3^2, occurring in position 520.
Sequence A228059 lists the subsequence of these numbers that are closer to being perfect than smaller numbers. - T. D. Noe, Aug 15 2013
Sequence A326137 lists terms with at least five distinct prime factors. See further comments there. - Antti Karttunen, Jun 13 2019

T. D. Noe, Table of n, a(n) for n = 1..10000
Charles Greathouse and Eric W. Weisstein, MathWorld: Odd perfect number
Oliver Knill, The oldest open problem in mathematics, Handout for NEU Math Circle, December 2, 2007
P. P. Nielsen, Odd Perfect Numbers Have At Least Nine Distinct Prime Factors, arXiv:math/0602485 [math.NT], 2006.
Wikipedia, Perfect number: Odd perfect numbers
Index entries for sequences where any odd perfect numbers must occur

Subsequence of A191218, and also of A228056 and A228057 (simpler versions of this sequence).
For various subsequences with additional conditions, see A228059, A325376, A325380, A325822, A326137 (with omega(n)>=5), A324898 (conjectured, subsequence if it does not contain any prime powers), A354362, A386425 (conjectured), A386427 (nondeficient terms), A386428 (powerful terms), A386429 U A351574.
Cf. A027748, A124010, A005408, A324647, A325319, A325320, A325375, A325377, A325378, A325379, A325819, A325823, A325824.

import Data.List (partition)
a228058 n = a228058_list !! (n-1)
a228058_list = filter f [1, 3 ..] where
   f x = length us == 1 && not (null vs) &&
         fst (head us) `mod` 4 == 1 && snd (head us) `mod` 4 == 1
         where (us,vs) = partition (odd . snd) $
                         zip (a027748_row x) (a124010_row x)
-- Reinhard Zumkeller, Aug 14 2013

nn = 100; n = 1; t = {}; While[Length[t] < nn, n = n + 2; {p, e} = Transpose[FactorInteger[n]]; od = Select[e, OddQ]; If[Length[e] > 1 && Length[od] == 1 && Mod[od[[1]], 4] == 1 && Mod[p[[Position[e, od[[1]]][[1,1]]]], 4] == 1, AppendTo[t, n]]]; t (* T. D. Noe, Aug 15 2013 *)

up_to = 1000;
isA228058(n) = if(!(n%2)||(omega(n)<2),0,my(f=factor(n),y=0); for(i=1,#f~,if(1==(f[i,2]%4), if((1==y)||(1!=(f[i,1]%4)),return(0),y=1), if(f[i,2]%2, return(0)))); (y));
A228058list(up_to) = { my(v=vector(up_to), k=0, n=0); while(kA228058(n), k++; v[k] = n)); (v); };
v228058 = A228058list(up_to);
A228058(n) = v228058[n]; \\ Antti Karttunen, Apr 22 2019

From Antti Karttunen, Apr 22 2019 & Jun 03 2019: (Start)
A325313(a(n)) = -A325319(n).
A325314(a(n)) = -A325320(n).
A001065(a(n)) = A325377(n).
A033879(a(n)) = A325379(n).
A034460(a(n)) = A325823(n).
A325814(a(n)) = A325824(n).
A324213(a(n)) = A325819(n).
(End)

Note in parentheses added to the definition by Antti Karttunen, Jun 03 2019

A191218 Odd numbers n such that sigma(n) is congruent to 2 modulo 4.

Original entry on oeis.org

5, 13, 17, 29, 37, 41, 45, 53, 61, 73, 89, 97, 101, 109, 113, 117, 137, 149, 153, 157, 173, 181, 193, 197, 229, 233, 241, 245, 257, 261, 269, 277, 281, 293, 313, 317, 325, 333, 337, 349, 353, 369, 373, 389, 397, 401, 405, 409, 421, 425, 433, 449, 457, 461, 477

Offset: 1

Luis H. Gallardo, May 26 2011

Exactly the numbers of the form p^{4k+1}*m^2 with p a prime congruent to 1 modulo 4 and m a positive integer coprime with p. The odd perfect numbers are all of this form.

See A228058 for the terms where m > 1. - Antti Karttunen, Apr 22 2019

			For n=3 one has a(3)=17 since sigma(17) = 18 = 4*4 +2 is congruent to 2 modulo 4

Antti Karttunen, Table of n, a(n) for n = 1..20000
Index entries for sequences related to sigma(n)

Subsequence of A191217.
Cf. A228058, A324898 (subsequences).
Cf. A000203, A191219, A324647, A324718, A324719.

with(numtheory): genodd := proc(b) local n,s,d; for n from 1 to b by 2 do s := sigma(n);
if modp(s,4)=2 then print(n); fi; od; end;

Select[Range[1,501,2],Mod[DivisorSigma[1,#],4]==2&] (* Harvey P. Dale, Nov 12 2017 *)

forstep(n=1,10^3,2,if(2==(sigma(n)%4),print1(n,", "))) \\ Joerg Arndt, May 27 2011

list(lim)=my(v=List()); forstep(e=1,logint(lim\=1,5),4, forprimestep(p=5,sqrtnint(lim,e),4, my(pe=p^e); forstep(m=1,sqrtint(lim\pe),2, if(m%p, listput(v,pe*m^2))))); Set(v) \\ Charles R Greathouse IV, Feb 16 2022

A325311 Odd abundant numbers k for which sigma(k) == 3 (mod 4).

Original entry on oeis.org

11025, 99225, 245025, 540225, 893025, 1334025, 2205225, 3980025, 4862025, 5832225, 6890625, 8037225, 8555625, 9828225, 10595025, 10989225, 12006225, 14402025, 19847025, 20385225, 24354225, 26163225, 26471025, 29648025, 31979025, 35820225, 38378025, 43758225, 46580625, 49491225, 50339025, 52490025, 55577025, 57836025, 60140025

Offset: 1

Antti Karttunen, Apr 20 2019

nonn

These are all squares. Square roots are in A325312.

Antti Karttunen, Table of n, a(n) for n = 1..10000
Index entries for sequences related to sigma(n).

Cf. A000203, A324647, A325312 (square roots).
Intersection of A005231 and A324899.
Subsequence of A156942.

Select[Range[1, 7755, 2]^2, Mod[(s = DivisorSigma[1, #]), 4] == 3 && s > 2*# &] (* Amiram Eldar, Apr 05 2024 *)

isA325311(n) = (n%2 && (3==sigma(n)%4) && sigma(n)>(2*n));

a(n) = A325312(n)^2. - Amiram Eldar, Apr 05 2024

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, ", ")))

A324727 Odd numbers such that 2n is equal to A318466(n), bitor(2*n,sigma(n)), where bitor is A003986.

Original entry on oeis.org

3, 7, 15, 21, 31, 55, 57, 63, 93, 105, 111, 127, 171, 189, 201, 213, 215, 217, 231, 237, 249, 253, 255, 315, 351, 357, 363, 369, 381, 393, 447, 465, 469, 473, 483, 489, 497, 501, 511, 651, 705, 747, 759, 789, 813, 831, 833, 879, 889, 895, 917, 959, 987, 989, 1001, 1015, 1023, 1155, 1365, 1377, 1407, 1467, 1491, 1503, 1505, 1515, 1533, 1595

Offset: 1

Antti Karttunen, Mar 15 2019

Odd terms of A324726.

Cf. A000203, A003986, A318466, A324647, A324718, A324719.

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

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

A324718 Odd numbers n for which bitand(2n,sigma(n)) = 2*bitand(n,sigma(n)-n), where bitand is bitwise-AND, A004198.

Original entry on oeis.org

1, 5, 9, 17, 37, 41, 73, 137, 149, 153, 257, 261, 277, 293, 337, 405, 521, 529, 549, 577, 593, 641, 661, 673, 677, 1025, 1033, 1061, 1093, 1097, 1109, 1153, 1193, 1289, 1297, 1301, 1321, 1361, 2053, 2069, 2081, 2089, 2097, 2113, 2129, 2209, 2213, 2225, 2309, 2341, 2377, 2389, 2593, 2633, 2689, 2693, 2729, 2825, 4129, 4133, 4177, 4229

Offset: 1

Antti Karttunen, Mar 14 2019

nonn

Odd numbers n for which 2*A318458(n) = A318468(n). If there are no common terms with A324719, then there are no odd perfect numbers.

This is not a subsequence of A191218, because terms 1, 9, 529, 2209, 10609, 77841, 83521, 263169, 279841, 330625, 528529, ... are not present in A191218.

Subsequence of A324639.

Cf. A004198, A191218, A318458, A318468, A324647, A324719, A324727.

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

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

A386425 Odd composites k such that sigma(k) has the same powerful part as k, where sigma is the sum of divisors function.

Original entry on oeis.org

153, 801, 1773, 3725, 4689, 4753, 5013, 6957, 8577, 8725, 9549, 9873, 11493, 13437, 14409, 15381, 18621, 19269, 21213, 21537, 23481, 25101, 26073, 26225, 28989, 29161, 29313, 29961, 32229, 33849, 34173, 36117, 38061, 39033, 40653, 42597, 43893, 47457, 47781, 48725, 48753, 51669, 52317, 54261, 56953, 57177, 57501

Offset: 1

Antti Karttunen, Aug 17 2025

nonn

By definition, the sequence contains all odd perfect numbers, and also includes any hypothetical odd triperfect number that is not a multiple of 3 (see A005820 and A347391), and similarly, any odd term of A046060 that is not a multiple of 5, etc. If there are no squares in this sequence (see conjecture in A386424), then the latter categories of numbers certainly do not exist, and this is then a subsequence of A228058.

The first nondeficient term is a(32315) = 81022725. See A386426.

Intersection of A071904 and A386424.
Nonsquare terms form a subsequence of A228058.
Cf. A000203, A003557, A057521, A386426 (nondeficient terms).
Cf. also A324647, A349749.

rad[n_] := Times @@ First /@ FactorInteger[n];a057521[n_] := n/Denominator[n/rad[n]^2];Select[Range[9,57501,2],!PrimeQ[#]&&a057521[DivisorSigma[1,#]]==a057521[#]&] (* James C. McMahon, Aug 18 2025 *)

A057521(n)=my(f=factor(n)); prod(i=1, #f~, if(f[i, 2]>1, f[i, 1]^f[i, 2], 1))
isA386425(n) = ((n>1) && (n%2) && !isprime(n) && (A057521(sigma(n))==A057521(n)));

{k | k is odd composite and A003557(A000203(k)) = A003557(k)}.

A324722 Numbers k such that A324658(A156552(k)) is zero.

Original entry on oeis.org

9, 21, 25, 35, 49, 55, 77, 95, 121, 125, 133, 143, 169, 185, 203, 209, 221, 265, 289, 299, 301, 319, 323, 343, 361, 371, 377, 413, 427, 437, 445, 451, 473, 481, 493, 497, 511, 527, 529, 531, 539, 553, 559, 583, 589, 605, 611, 623, 629, 667, 679, 689, 703, 707, 737, 763, 767, 779, 791, 793, 799, 805, 817, 841, 845, 847, 851, 869, 871, 899, 901

Offset: 1

Antti Karttunen, Mar 15 2019

nonn

First even term is A005940(1+A324647(1)) = A005940(1+1116225) = 1912898. - Typo corrected by Antti Karttunen, Jul 21 2021

Antti Karttunen, Table of n, a(n) for n = 1..694 (based on Hans Havermann's factorization of A156552)
Index entries for sequences related to binary expansion of n
Index entries for sequences computed from indices in prime factorization
Index entries for sequences related to sigma(n)

Cf. A005940, A156552, A324647, A324652, A324658, A324723.

Positions of zeros in A324716.

isA324652(n) = ((2*n)==bitand(2*n, sigma(n)));
isA324722(n) = if(n>1,isA324652(A156552(n)));
k=0; n=0; while(k<105, n++; if(isA324722(n), k++; 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,", ")));

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

A228058 Odd numbers of the form p^(1+4k) * r^2, where p is prime of the form 1+4m, r > 1, and gcd(p,r) = 1. (Euler's criteria for odd perfect numbers).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Haskell

Mathematica

PARI

Formula

Extensions

A191218 Odd numbers n such that sigma(n) is congruent to 2 modulo 4.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

PARI

A325311 Odd abundant numbers k for which sigma(k) == 3 (mod 4).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

A324727 Odd numbers such that 2n is equal to A318466(n), bitor(2*n,sigma(n)), where bitor is A003986.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

A324718 Odd numbers n for which bitand(2n,sigma(n)) = 2*bitand(n,sigma(n)-n), where bitand is bitwise-AND, A004198.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

A386425 Odd composites k such that sigma(k) has the same powerful part as k, where sigma is the sum of divisors function.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A324722 Numbers k such that A324658(A156552(k)) is zero.

Views

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