cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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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).

Original entry on oeis.org

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

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Author

T. D. Noe, Aug 13 2013

Keywords

Comments

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

Links

Crossrefs

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.

Programs

  • Haskell
    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
  • Mathematica
    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 *)
  • PARI
    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

Formula

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)

Extensions

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

A348743 Odd nonsquares k for which A161942(k) >= k, where A161942 is the odd part of sigma.

Original entry on oeis.org

2205, 19845, 108045, 143325, 178605, 187425, 236925, 266805, 319725, 353925, 372645, 407925, 452025, 462825, 584325, 637245, 646425, 658125, 672525, 789525, 796005, 804825, 845325, 920205, 972405, 981225, 1007325, 1055925, 1069425, 1102725, 1113525, 1116225, 1166445, 1201725, 1245825, 1289925, 1378125, 1380825, 1442925
Offset: 1

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Author

Antti Karttunen, Nov 02 2021

Keywords

Comments

The first non-multiples of 5 are a(103) = 6243237 and a(125) = 8164233.
From Antti Karttunen, Nov 28 2024: (Start)
This is not a subsequence of A228058. At least k = A000040(28)*(A002110(27)/2)^2 = 15388519572341080054329140040512468358441210638435506649120749687401476705908239675 is a number of the form 4m+3 such that A161942(k) >= k.
Another such number is A000040(28)*81*(A002110(25)/6)^2 = 1279741205456530915782536871495922949062895982530933679752838870798129159675.
Question: What is the smallest term of this sequence that is of the form 4m+3, and thus not in A386427 (in A191218 and in A228058)?
(End)

Links

Crossrefs

Intersection of A088828 and A348742.
Cf. A386427 (a subsequence, which agrees for a very long time).
Cf. A033630, A065205, A083206, A161942, A191218, A228058, A336702, A348741.
Cf. also A065235, A162284.

Programs

  • PARI
    A000265(n) = (n >> valuation(n, 2));
isA348743(n) = ((n%2)&&!issquare(n)&&A000265(sigma(n))>=n); \\ Edited Nov 28 2024

Extensions

Definition changed (from > to >=) to formally include also any hypothetical odd perfect numbers - Antti Karttunen, Nov 28 2024
Comment removed, because it was more related to sequence A386427. - Antti Karttunen, Aug 21 2025

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

Views

Author

Antti Karttunen, Aug 17 2025

Keywords

Comments

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.

Links

Crossrefs

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

Programs

  • PARI
    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))));

Formula

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

Extensions

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