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 A065403 Primes of the form sigma(m^2) where m is a composite number ordered by values m. 9
 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 (list; graph; refs; listen; history; text; internal format)
 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]) } \\ 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

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Last modified April 16 18:53 EDT 2021. Contains 343050 sequences. (Running on oeis4.)