cp's OEIS Frontend

A065403 Primes of the form sigma(m^2) where m is a composite number ordered by values m.

%I A065403 #31 Aug 18 2024 17:28:39
%S A065403 31,127,1093,2801,8191,19531,30941,131071,88741,524287,292561,797161,
%T A065403 732541,3500201,5229043,12207031,25646167,28792661,39449441,48037081,
%U A065403 305175781,262209281,917087137,2147483647,1394714501,2666986681
%N A065403 Primes of the form sigma(m^2) where m is a composite number ordered by values m.
%C A065403 There are 46 cases below 10^12.
%C A065403 All Mersenne primes are here: sigma((2^((p-1)/2))^2) = sigma(2^(p-1)) = -1 + 2^p, for suitable p.
%C A065403 m is of the form p^(2*e) for some prime p and e > 1 as sigma is multiplicative and m is composite. Terms are sorted by values of m. The sequence isn't monotonic. - _David A. Corneth_, Jul 18 2020
%H A065403 David A. Corneth, <a href="/A065403/b065403.txt">Table of n, a(n) for n = 1..10000</a> (first 100 terms from Harry J. Smith, terms 101-500 from Amiram Eldar)
%e A065403 19531 is in the sequence as for the composite m = 125 we have sigma(m^2) = 19531. - _David A. Corneth_, Jul 18 2020
%t A065403 Do[s=DivisorSigma[1, n]; If[PrimeQ[s]&&!PrimeQ[Sqrt[n]], Print[{n, Sqrt[n], s}]], {n, 1, 20000000}]
%o A065403 (PARI) { n=0; for (m=1, 10^9, if (isprime(m), next); x=sigma(m^2); if (isprime(x), write("b065403.txt", n++, " ", x); if (n==100, return)) ) } \\ _Harry J. Smith_, Oct 18 2009
%o A065403 (PARI) upto(n) = {res = List(); forstep(e = 4, logint(n, 2), 2, forprime(p = 2, sqrtnint(n, e), c = (p^(e + 1) - 1)/(p - 1); if(isprime(c), listput(res, [p^e, c]) ) ) ); listsort(res); vector(#res, i, res[i][2]) } \\ _David A. Corneth_, Jul 18 2020
%Y A065403 Cf. A062700, A000203, A001348, A065404, A065405.
%K A065403 nonn
%O A065403 1,1
%A A065403 _Labos Elemer_, Nov 06 2001
%E A065403 Name corrected by _David A. Corneth_, Jul 18 2020