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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).

%I A228058 #44 Aug 21 2025 14:36:08
%S A228058 45,117,153,245,261,325,333,369,405,425,477,549,605,637,657,725,801,
%T A228058 833,845,873,909,925,981,1017,1025,1053,1233,1325,1341,1377,1413,1421,
%U A228058 1445,1525,1557,1573,1629,1737,1773,1805,1813,1825,2009,2057,2061,2097,2169
%N 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).
%C A228058 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.
%C A228058 Sequence A228059 lists the subsequence of these numbers that are closer to being perfect than smaller numbers. - _T. D. Noe_, Aug 15 2013
%C A228058 Sequence A326137 lists terms with at least five distinct prime factors. See further comments there. - _Antti Karttunen_, Jun 13 2019
%H A228058 T. D. Noe, <a href="/A228058/b228058.txt">Table of n, a(n) for n = 1..10000</a>
%H A228058 Charles Greathouse and Eric W. Weisstein, <a href="https://mathworld.wolfram.com/OddPerfectNumber.html">MathWorld: Odd perfect number</a>
%H A228058 Oliver Knill, <a href="http://www.math.harvard.edu/~knill/seminars/perfect/handout.pdf">The oldest open problem in mathematics</a>, Handout for NEU Math Circle, December 2, 2007
%H A228058 P. P. Nielsen, <a href="http://arxiv.org/abs/math/0602485">Odd Perfect Numbers Have At Least Nine Distinct Prime Factors</a>, arXiv:math/0602485 [math.NT], 2006.
%H A228058 Wikipedia, <a href="https://en.wikipedia.org/wiki/Perfect_number#Odd_perfect_numbers">Perfect number: Odd perfect numbers</a>
%H A228058 <a href="/index/O#opnseqs">Index entries for sequences where any odd perfect numbers must occur</a>
%F A228058 From _Antti Karttunen_, Apr 22 2019 & Jun 03 2019: (Start)
%F A228058 A325313(a(n)) = -A325319(n).
%F A228058 A325314(a(n)) = -A325320(n).
%F A228058 A001065(a(n)) = A325377(n).
%F A228058 A033879(a(n)) = A325379(n).
%F A228058 A034460(a(n)) = A325823(n).
%F A228058 A325814(a(n)) = A325824(n).
%F A228058 A324213(a(n)) = A325819(n).
%F A228058 (End)
%t A228058 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 *)
%o A228058 (Haskell)
%o A228058 import Data.List (partition)
%o A228058 a228058 n = a228058_list !! (n-1)
%o A228058 a228058_list = filter f [1, 3 ..] where
%o A228058    f x = length us == 1 && not (null vs) &&
%o A228058          fst (head us) `mod` 4 == 1 && snd (head us) `mod` 4 == 1
%o A228058          where (us,vs) = partition (odd . snd) $
%o A228058                          zip (a027748_row x) (a124010_row x)
%o A228058 -- _Reinhard Zumkeller_, Aug 14 2013
%o A228058 (PARI)
%o A228058 up_to = 1000;
%o A228058 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));
%o A228058 A228058list(up_to) = { my(v=vector(up_to), k=0, n=0); while(k<up_to, n++; if(isA228058(n), k++; v[k] = n)); (v); };
%o A228058 v228058 = A228058list(up_to);
%o A228058 A228058(n) = v228058[n]; \\ _Antti Karttunen_, Apr 22 2019
%Y A228058 Subsequence of A191218, and also of A228056 and A228057 (simpler versions of this sequence).
%Y A228058 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.
%Y A228058 Cf. A027748, A124010, A005408, A324647, A325319, A325320, A325375, A325377, A325378, A325379, A325819, A325823, A325824.
%K A228058 nonn,changed
%O A228058 1,1
%A A228058 _T. D. Noe_, Aug 13 2013
%E A228058 Note in parentheses added to the definition by _Antti Karttunen_, Jun 03 2019