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A228059 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 that are closer to being perfect than previous terms. 5
45, 405, 2205, 26325, 236925, 1380825, 1660725, 35698725, 3138290325, 29891138805, 73846750725, 194401220013, 194509436121, 194581580193, 194689796301, 194798012409, 194906228517, 194942300553, 195230876841, 195339092949, 195447309057, 195699813309 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

A number x is perfect if sigma(x) = 2x, where sigma is the sum of divisors of x. See A228058 for numbers of the form p^(1+4k) * r^2. This sequence ends when the first odd perfect number occurs.

The first two papers by Dris listed below are for information only; this sequence in independent of the papers. In the second paper, Dris attempts to prove that the exponent of p above is 1 for odd perfect numbers. Coincidently, the first 9 numbers in this sequence have exponent 1.

a(38) > 10^12. - Giovanni Resta, Aug 16 2018

a(38) <= 283665529390725 = 15349 * (3^3 * 5 * 19 * 53)^2. - Giovanni Resta, Aug 23 2018

LINKS

Giovanni Resta, Table of n, a(n) for n = 1..37

Jose Arnaldo B. Dris, The abundancy index of divisors of odd perfect numbers, J. Integer Sequences, 15 (2012), Article 12.4.4

Jose Arnaldo B. Dris, A short "proof" for Sorli's conjecture on odd perfect numbers, arxiv 1308.2156 [math.NT], 2013-2015.

Jose Arnaldo B. Dris, Euclid-Euler Heuristics for (Odd) Perfect Numbers, arXiv preprint arXiv:1310.5616 [math.NT], 2013-2017.

Jose Arnaldo B. Dris, A Sufficient Condition for Disproving Descartes's Conjecture on Odd Perfect Numbers, arXiv preprint arXiv:1311.6803 [math.NT], 2013-2015.

Jose Arnaldo Bebita Dris, Doli-Jane Uvales Tejada, A note on the OEIS sequence A228059, Notes on Number Theory and Discrete Mathematics (2019) Vol. 25, No. 1, 199-205.

Index entries for sequences where any odd perfect numbers must occur

EXAMPLE

           45 =   5 * 3^2.

          405 =   5 * 3^4.

         2205 =   5 * (3 * 7)^2.

        26325 =  13 * (3^2 * 5)^2.

       236925 =  13 * (3^3 * 5)^2.

      1380825 =  17 * (3 * 5 * 19)^2.

      1660725 =  61 * (3 * 5 * 11)^2.

     35698725 =  61 * (3^2 * 5 * 17)^2.

   3138290325 =  53 * (3^4 * 5 * 19)^2.

  29891138805 =   5 * (3^2 * 11^2 * 71)^2.

  73846750725 = 509 * (3 * 5 * 11 * 73)^2.

MATHEMATICA

nn = 7; f[n_] := Abs[DivisorSigma[1, n]/n - 2]; n = 45; t = {n}; lastF = f[n]; cnt = 1; While[cnt < 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 && f[n] < lastF, cnt++; lastF = f[n]; Print[{n, lastF}]; AppendTo[t, n]]]; t

PROG

(PARI)

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));

m=-1; n=0; while(m!=0, n++; if(isA228058(n), if((m<0) || abs((sigma(n)/n)-2)<m, m=abs((sigma(n)/n)-2); print1(n, ", ")))); \\ Antti Karttunen, Apr 22 2019

CROSSREFS

Cf. A000203 (sigma), A000396 (perfect numbers), A228058, A325379.

Sequence in context: A272850 A129153 A156719 * A155015 A179795 A036495

Adjacent sequences:  A228056 A228057 A228058 * A228060 A228061 A228062

KEYWORD

nonn

AUTHOR

T. D. Noe, Aug 14 2013

EXTENSIONS

a(10) (as communicated by T. D. Noe) added by Jose Arnaldo Bebita Dris, Aug 16 2018

a(11)-a(22) from Giovanni Resta, Aug 16 2018

STATUS

approved

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Last modified September 26 23:42 EDT 2021. Contains 347673 sequences. (Running on oeis4.)