cp's OEIS Frontend

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.

%I A228059 #78 Aug 05 2025 10:04:41
%S A228059 45,405,2205,26325,236925,1380825,1660725,35698725,3138290325,
%T A228059 29891138805,73846750725,194401220013,194509436121,194581580193,
%U A228059 194689796301,194798012409,194906228517,194942300553,195230876841,195339092949,195447309057,195699813309
%N 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.
%C A228059 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.
%C A228059 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.
%C A228059 a(38) > 10^12. - _Giovanni Resta_, Aug 16 2018
%C A228059 a(38) <= 283665529390725 = 15349 * (3^3 * 5 * 19 * 53)^2. - _Giovanni Resta_, Aug 23 2018
%C A228059 a(39) <= 3116918388785625 = 37993 * (3^2 * 5^2 * 19 * 67)^2. - _Alexander Violette_, Mar 05 2022
%C A228059 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
%H A228059 Giovanni Resta, <a href="/A228059/b228059.txt">Table of n, a(n) for n = 1..37</a>
%H A228059 Jose Arnaldo B. Dris, <a href="http://cs.uwaterloo.ca/journals/JIS/VOL15/Dris/dris8.html">The abundancy index of divisors of odd perfect numbers</a>, J. Integer Sequences, 15 (2012), Article 12.4.4.
%H A228059 Jose Arnaldo B. Dris, <a href="http://arxiv.org/abs/1308.2156">A short "proof" for Sorli's conjecture on odd perfect numbers</a>, arxiv 1308.2156 [math.NT], 2013-2015.
%H A228059 Jose Arnaldo B. Dris, <a href="http://arxiv.org/abs/1310.5616">Euclid-Euler Heuristics for (Odd) Perfect Numbers</a>, arXiv preprint arXiv:1310.5616 [math.NT], 2013-2017.
%H A228059 Jose Arnaldo B. Dris, <a href="http://arxiv.org/abs/1311.6803">A Sufficient Condition for Disproving Descartes's Conjecture on Odd Perfect Numbers</a>, arXiv preprint arXiv:1311.6803 [math.NT], 2013-2015.
%H A228059 Jose Arnaldo Bebita Dris, Doli-Jane Uvales Tejada, <a href="https://doi.org/10.7546/nntdm.2019.25.1.199-205">A note on the OEIS sequence A228059</a>, Notes on Number Theory and Discrete Mathematics (2019) Vol. 25, No. 1, 199-205.
%H A228059 <a href="/index/O#opnseqs">Index entries for sequences where any odd perfect numbers must occur</a>
%e A228059            45 =   5 * 3^2.
%e A228059           405 =   5 * 3^4.
%e A228059          2205 =   5 * (3 * 7)^2.
%e A228059         26325 =  13 * (3^2 * 5)^2.
%e A228059        236925 =  13 * (3^3 * 5)^2.
%e A228059       1380825 =  17 * (3 * 5 * 19)^2.
%e A228059       1660725 =  61 * (3 * 5 * 11)^2.
%e A228059      35698725 =  61 * (3^2 * 5 * 17)^2.
%e A228059    3138290325 =  53 * (3^4 * 5 * 19)^2.
%e A228059   29891138805 =   5 * (3^2 * 11^2 * 71)^2.
%e A228059   73846750725 = 509 * (3 * 5 * 11 * 73)^2.
%t A228059 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
%o A228059 (PARI)
%o A228059 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 A228059 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
%Y A228059 Cf. A000203 (sigma), A000396 (perfect numbers), A228058, A325379, A349752.
%Y A228059 Cf. also A171929.
%K A228059 nonn
%O A228059 1,1
%A A228059 _T. D. Noe_, Aug 14 2013
%E A228059 a(10) (as communicated by _T. D. Noe_) added by _Jose Arnaldo Bebita Dris_, Aug 16 2018
%E A228059 a(11)-a(22) from _Giovanni Resta_, Aug 16 2018