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

31, 127, 1093, 2801, 8191, 19531, 30941, 131071, 88741, 524287, 292561, 797161, 732541, 3500201, 5229043, 12207031, 25646167, 28792661, 39449441, 48037081, 305175781, 262209281, 917087137, 2147483647, 1394714501, 2666986681

Offset

1,1

Comments

There are 46 cases below 10^12.

All Mersenne primes are here: sigma((2^((p-1)/2))^2) = sigma((2^(p-1)) = -1 + 2^p, for suitable p.

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

Links

David A. Corneth, Table of n, a(n) for n = 1..10000 (first 100 terms from Harry J. Smith, terms 101-500 from Amiram Eldar)

Example

19531 is in the sequence as for the composite m = 125 we have sigma(m^2) = 19531. - David A. Corneth, Jul 18 2020

Mathematica

Do[s=DivisorSigma[1, n]; If[PrimeQ[s]&&!PrimeQ[Sqrt[n]], Print[{n, Sqrt[n], s}]], {n, 1, 20000000}]

Prog

(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
(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

Crossrefs

Cf. A062700, A000203, A001348, A065404, A065405.

Sequence in context: A158563 A079141 A049203 * A278054 A035502 A259509

Adjacent sequences: A065400 A065401 A065402 * A065404 A065405 A065406

Keyword

nonn

Author

Labos Elemer, Nov 06 2001

Extensions

Name corrected by David A. Corneth, Jul 18 2020

Status

approved