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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). 51
 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 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 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 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) 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[], 4] == 1 && Mod[p[[Position[e, od[]][[1, 1]]]], 4] == 1, AppendTo[t, n]]]; t (* T. D. Noe, Aug 15 2013 *) PROG (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 (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(k

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Last modified May 15 01:50 EDT 2021. Contains 343909 sequences. (Running on oeis4.)