A386425 -id:A386425 - 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

A386424 Numbers 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

1, 2, 5, 12, 13, 26, 29, 37, 41, 44, 56, 61, 73, 74, 76, 90, 101, 109, 113, 122, 137, 146, 153, 157, 172, 173, 181, 193, 218, 229, 236, 257, 268, 277, 281, 312, 313, 314, 317, 353, 362, 373, 386, 389, 397, 401, 409, 421, 433, 457, 458, 461, 509, 522, 524, 528, 541, 554, 560, 569, 601, 613, 617, 626, 641, 652, 653

Offset: 1

Antti Karttunen, Aug 17 2025

Conjecture 1: the initial 1 is the only square in this sequence, and a(2) = 2 is the only term that is twice a square.

Conjecture 2: A323653 is a subsequence (which would follow from conjecture 1 (c) given there).

Cf. A000203, A003557, A057521, A387156.
Subsequences: A323653 (conjectured), A351549, A386425 (odd composites), A386426 (nondeficient terms).
Cf. also A006872, A351446, A387158.

rad[n_] := Times @@ First /@ FactorInteger[n];a057521[n_] := n/Denominator[n/rad[n]^2];Select[Range[653],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))
isA386424(n) = (A057521(sigma(n))==A057521(n));

{k | A057521(A000203(k)) = A057521(k)}, or equally, {k | A387156(k) = A003557(k)}.

A386426 Odd nondeficient numbers 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

81022725, 891360225, 992106225, 1863765225, 2349967725, 3322372725, 7211992725, 8670600225, 9156802725, 11101612725, 13208490225, 15477435225, 15963637725, 18394650225, 18880852725, 21311865225, 21960135225, 22446337725, 22932540225, 25687687725, 25849755225, 28280767725, 28604902725, 30711780225, 31035915225

Offset: 1

Antti Karttunen, Aug 17 2025

Sequence by definition contains also any such hypothetical odd terms of A007691 that are mentioned in the comments of A386425. However, if no such terms exist, then this is a subsequence of A386427.
This sequence contains also the intersection of A001694 and A386425, even though it is probably an empty set. See comments in A386428.
The first three terms not divisible by 25 are: a(191) = 283806508293, a(247) = 371184932349, a(328) = 502252568433.

Intersection of A023196 and A386425.
Conjectured to be a subsequence of A386427.
Cf. A000203, A001694, A003557, A007691, A057521, A386428.
Cf. also A005231.

A057521(n)=my(f=factor(n)); prod(i=1, #f~, if(f[i, 2]>1, f[i, 1]^f[i, 2], 1))
isA386426(n) = if(!(n%2), 0, my(s=sigma(n)); ((s>=(2*n)) && (A057521(s)==A057521(n))));

{k | k is odd, A000203(k) >= 2*k and A003557(A000203(k)) = A003557(k)}.

a(8)-a(25) from Giovanni Resta, Aug 18 2025

A387156 a(n) = A003557(sigma(n)), where A003557(n) is multiplicative with a(p^e) = p^(e-1), and sigma is the sum of divisors function.

Original entry on oeis.org

1, 1, 2, 1, 1, 2, 4, 1, 1, 3, 2, 2, 1, 4, 4, 1, 3, 1, 2, 1, 16, 6, 4, 2, 1, 1, 4, 4, 1, 12, 16, 3, 8, 9, 8, 1, 1, 2, 4, 3, 1, 16, 2, 2, 1, 12, 8, 2, 1, 1, 12, 7, 9, 4, 12, 4, 8, 3, 2, 4, 1, 16, 4, 1, 2, 24, 2, 3, 16, 24, 12, 1, 1, 1, 2, 2, 16, 4, 8, 1, 11, 3, 2, 16, 18, 2, 4, 6, 3, 3, 8, 4, 64, 24, 4, 6, 7, 3, 2

Offset: 1

Antti Karttunen, Aug 19 2025

nonn

Antti Karttunen, Table of n, a(n) for n = 1..10000

Cf. A000203, A003557, A062401, A080398, A387157.

Cf. also A386424, A386425.

A387156[n_] := # / Times @@ FactorInteger[#][[All, 1]] & [DivisorSigma[1, n]];
Array[A387156, 100] (* Paolo Xausa, Aug 20 2025 *)

A387156(n) = { my(s=sigma(n)); s/factorback(factor(s)[,1]); };

a(n) = A000203(n) / A080398(n).

a(n) = A062401(n) / A387157(n).

A387159 Odd numbers k such that A173557(k) = A173557(sigma(k)), where A173557(n) is multiplicative with a(p^e) = p-1 and sigma is the sum of divisors function.

Original entry on oeis.org

1, 63, 135, 351, 875, 891, 999, 1647, 1859, 1971, 4239, 5211, 7479, 8451, 10719, 11367, 12339, 14607, 16317, 16551, 17847, 18171, 19791, 20439, 22103, 23679, 26919, 27951, 29511, 31131, 31407, 31487, 32427, 32751, 33399, 35667, 37287, 39231, 43767, 44739, 47331, 50571, 52191, 53811, 54459, 57319, 57699, 63207, 66771

Offset: 1

Antti Karttunen, Aug 19 2025

nonn

Odd numbers k for which A173557(k) == A387157(k).

Odd terms of A387158.
Cf. A000203, A173557, A387157.
Cf. also A351443, A353679, A386425.

A387159Q[k_] := OddQ[k] && #[k] == #[DivisorSigma[1, k]] & [Times @@ (FactorInteger[#][[All, 1]] - 1) &];
Select[Range[100000], A387159Q] (* Paolo Xausa, Aug 20 2025 *)

A173557(n) = factorback(apply(p -> p-1,factor(n)[,1]));
is_A387159(n) = (n%2 && (A173557(sigma(n))==A173557(n)));

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).

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A386424 Numbers k such that sigma(k) has the same powerful part as k, where sigma is the sum of divisors function.

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A386426 Odd nondeficient numbers k such that sigma(k) has the same powerful part as k, where sigma is the sum of divisors function.

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A387156 a(n) = A003557(sigma(n)), where A003557(n) is multiplicative with a(p^e) = p^(e-1), and sigma is the sum of divisors function.

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A387159 Odd numbers k such that A173557(k) = A173557(sigma(k)), where A173557(n) is multiplicative with a(p^e) = p-1 and sigma is the sum of divisors function.

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