A386429 -id:A386429 - 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

A347874 Odd composites for which A342926(n) is even and A342926(2*n) is a multiple of 3.

Original entry on oeis.org

45, 153, 261, 325, 369, 405, 441, 477, 801, 909, 925, 1017, 1233, 1341, 1377, 1521, 1525, 1557, 1573, 1773, 1825, 2097, 2205, 2313, 2349, 2401, 2421, 2425, 2529, 2637, 2725, 2853, 3177, 3249, 3321, 3501, 3609, 3645, 3757, 3825, 3925, 4041, 4149, 4293, 4477, 4525, 4581, 4689, 4825, 5013, 5121, 5337, 5445, 5553, 5725

Offset: 1

Antti Karttunen, Sep 18 2021

Numbers k for which A347871(k) = 0 and A347883(2*k) = 0.

This is not a subsequence of A228058. The terms that do not occur there: 441, 1521, 2401, 3249, 8649, 16641, 28561, 35721, etc., seem all to be squares. Terms of A228058 missing from this sequence are: 117, 245, 333, 425, 549, 605, 637, 657, 725, etc. (See A351574.) Terms that are also in A228058 are given in A386429.

Cf. A000203, A003415, A228058, A342925, A342926, A347871, A347883, A351574, A386429 (subsequence).

Intersection of A347872 and A351562.

ad[1] = 0; ad[n_] := n * Total@(Last[#]/First[#]& /@ FactorInteger[n]); f[n_] := ad[DivisorSigma[1, n]] - n; Select[Range[1, 5725, 2], CompositeQ[#] && EvenQ[f[#]] && Divisible[f[2*#], 3] &] (* Amiram Eldar, Sep 18 2021 *)

A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A342926(n) = (A003415(sigma(n))-n);
isA347874(n) = ((n%2)&&!isprime(n)&&!(A342926(n)%2)&&!(A342926(2*n)%3));

A351574 Terms of A228058 missing from A347874.

Original entry on oeis.org

117, 245, 333, 425, 549, 605, 637, 657, 725, 833, 845, 873, 981, 1025, 1053, 1325, 1413, 1421, 1445, 1629, 1737, 1805, 1813, 2009, 2057, 2061, 2169, 2225, 2493, 2525, 2597, 2645, 2817, 2825, 2873, 2925, 2989, 2997, 3033, 3141, 3357, 3425, 3509, 3573, 3577, 3681, 3725, 3789, 3897, 4113, 4205, 4325, 4361, 4693, 4753

Offset: 1

Antti Karttunen, Feb 23 2022

Numbers that satisfy Euler's criterion for the odd perfect numbers (A228058), but do not satisfy the criterion specified in A347874.

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

Setwise difference A228058 \ A347874, also setwise difference A228058 \ A386429.

Cf. A000203, A003415.

A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A342926(n) = (A003415(sigma(n))-n);
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));
isA347874(n) = ((n%2)&&!isprime(n)&&!(A342926(n)%2)&&!(A342926(2*n)%3));
isA351574(n) = (isA228058(n) && !isA347874(n));

A387162 Numbers k satisfying Euler's criterion for odd perfect numbers (A228058), such that sigma(k)+k is also a multiple of 3, and sigma(k) preserves the 3-adic valuation of k, where sigma is the sum of divisors function.

Original entry on oeis.org

153, 325, 801, 925, 1525, 1573, 1773, 1825, 2097, 2205, 2425, 2725, 3757, 3825, 3925, 4041, 4477, 4525, 4689, 4825, 5013, 5725, 6025, 6877, 6925, 6957, 7381, 7605, 7825, 7929, 8125, 8425, 8577, 8725, 8833, 9325, 9549, 9873, 9925, 10225, 10525, 10693, 10825, 10933, 11425, 11493, 11737, 12789, 13189, 13437, 13525

Offset: 1

Antti Karttunen, Aug 27 2025

Intersection of A228058 and A349752.
Subsequence of A349755 from which this differs for the first time at n=109, with a(109) = 31225, while A349755(109) = 31213.
Probably the intersection of A349755 and A386429.

nn=275;n=1;a228058={};While[Length[a228058 ] < 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[a228058,n]]];lim=a228058[[-1]];a349752=Select[Range[1,lim,2],Divisible[(s=DivisorSigma[1,#])+#,3] && IntegerExponent[s,3]==IntegerExponent[#,3]&];Intersection[a228058,a349752] (* James C. McMahon, Aug 27 2025 *)

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));
isA349752(n) = if(!(n%2), 0, my(s=sigma(n)); (0==(s+n)%3) && valuation(s, 3)==valuation(n, 3));
isA387162(n) = (isA349752(n) && isA228058(n));

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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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A347874 Odd composites for which A342926(n) is even and A342926(2*n) is a multiple of 3.

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A351574 Terms of A228058 missing from A347874.

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A387162 Numbers k satisfying Euler's criterion for odd perfect numbers (A228058), such that sigma(k)+k is also a multiple of 3, and sigma(k) preserves the 3-adic valuation of k, where sigma is the sum of divisors function.

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