A063783 Numbers k such that the sum of the cubes of divisors of k is a prime.
4, 9, 121, 36481, 72361, 146689, 259081, 654481, 683929, 786769, 1985281, 2036329, 3193369, 3636649, 3798601, 4583881, 5031049, 5470921, 5555449, 6135529, 6713281, 7284601, 7778521, 16589329, 20403289, 21557449, 22915369, 26739241, 27426169, 30261001, 30591961
Offset: 1
Keywords
Examples
All these terms are squares of primes {2, 3, 11, 191, 269, 383, 509, 809, 827, 887, 1409, 1427, 1787, 1907, 1949, 2141, 2243, 2339, 2357, 2477, 2591, 2699, 2789, ...}, so their sigma_3(p^2) = p^6 + p^3 + 1 has polynomial of degree 6.
sigma_3(9) = 1 + 27 + 729 = 757 is a prime.
Links
- Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Harry J. Smith)
Programs
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Mathematica
Select[Prime[Range[500]]^2, PrimeQ@ DivisorSigma[3, #] &] (* Michael De Vlieger, Jul 16 2017 *)
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PARI
{ n=0; p=0; for (m=1, 10^9, p=nextprime(p+1); if(isprime(p^6 + p^3 + 1), write("b063783.txt", n++, " ", p^2); if (n==1000, break)) ) } \\ Harry J. Smith, Aug 31 2009
Formula
a(n) = A066100(n)^2. - Amiram Eldar, Aug 16 2024
Comments