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.

45, 405, 2205, 26325, 236925, 1380825, 1660725, 35698725, 3138290325, 29891138805, 73846750725, 194401220013, 194509436121, 194581580193, 194689796301, 194798012409, 194906228517, 194942300553, 195230876841, 195339092949, 195447309057, 195699813309

Offset: 1

Author

T. D. Noe, Aug 14 2013

Keywords

• nonn

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

a(39) <= 3116918388785625 = 37993 * (3^2 * 5^2 * 19 * 67)^2. - Alexander Violette, Mar 05 2022

The first 37 terms are all multiples of 3, as well as the two additional terms given above. See also comments in A349752. - Antti Karttunen, Jan 04 2025

Examples

			           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.
		

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

Crossrefs

Cf. A000203 (sigma), A000396 (perfect numbers), A228058, A325379, A349752.

Cf. also A171929.

Programs

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

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)Antti Karttunen, Apr 22 2019

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