A324898 -id:A324898 - 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

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

Original entry on oeis.org

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

A324649 Numbers k such that A318458(k) (bitwise-AND of k and sigma(k)-k) is equal to k.

Original entry on oeis.org

6, 20, 28, 36, 66, 72, 88, 100, 104, 114, 132, 150, 240, 258, 264, 272, 280, 304, 354, 368, 392, 402, 464, 496, 498, 516, 550, 552, 642, 644, 680, 708, 748, 770, 774, 784, 786, 834, 836, 840, 860, 978, 1026, 1032, 1040, 1044, 1056, 1062, 1064, 1068, 1074, 1092, 1104, 1120, 1184, 1232, 1266, 1312, 1362, 1376, 1410, 1504

Offset: 1

Antti Karttunen, Mar 14 2019

nonn

Positions of zeros in A324648. Fixed points of A318458, also positions of the records in the latter.
Intersection with A324652 gives A324643.
The odd terms are: 7425, 76545, 92565, ... (A324897).

Cf. A318458, A324648, A324652.

Cf. A000396, A324643, A324897, A324898 (subsequences).

Select[Range@ 1600, BitAnd[#, DivisorSigma[1, #] - #] == # &] (* Michael De Vlieger, Apr 21 2019, after Vincenzo Librandi at A318458 *)

for(n=1,oo,if(bitand(n,sigma(n)-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

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

Formula

Extensions

A324649 Numbers k such that A318458(k) (bitwise-AND of k and sigma(k)-k) is equal to k.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

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