A279096 Numbers k such that sigma(k^3) is prime.

4, 9, 16, 25, 64, 81, 169, 289, 625, 961, 1024, 2401, 3721, 5329, 7921, 22201, 26569, 63001, 121801, 124609, 212521, 273529, 358801, 418609, 744769, 885481, 896809, 1048576, 1181569, 1247689, 1510441, 1630729, 1666681, 1682209, 1771561, 1874161, 1985281

Offset: 1

Author

Jon E. Schoenfield, Mar 12 2017

Keywords

• nonn

Comments

All terms are square. Moreover, each term is of the form p^j where both p and j*3 + 1 are prime (see A279094).

Examples

			4 is in the sequence because sigma(4^3) = sigma(2^6) = 1 + 2 + 4 + 8 + 16 + 32 + 64 = 127, which is prime.
16 is in the sequence because sigma(16^3) = sigma(2^12) = Sum_{m=0..12} 2^m = (2^13 - 1)/(2 - 1) = 8191, which is prime.
36 is not in the sequence because sigma(36^3) = sigma(2^6*3^6) = ((2^7 - 1)/(2 - 1))*((3^7 - 1)/(3 - 1)) = 127*1093, which is not prime. (36 is not of the form p^j where p is prime.)
361 is not in the sequence (even though 361 = 19^2 is of the form p^j where both p and 3*j + 1 are prime) because sigma(361^3) = sigma(19^6) = (19^7 - 1)/(19 - 1) = 49659541 = 701 * 70841.
		

Links

• Giovanni Resta, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000203 (sigma(k)), A023194 (sigma(k) is prime), A055638 (sigma(k^2) is prime), A279094 (smallest k such that sigma(k^n) is prime).

Programs

Mathematica

mx = 10^7; ee = Select[Range@ Log2@ mx, PrimeQ[3 # + 1] &]; Union@ Reap[ Do[ Do[ If[(v = p^e) <= mx, If[ PrimeQ[(p v^3 - 1)/ (p-1)], Sow@ v], Break[]], {e, ee}], {p, Prime@ Range@ PrimePi@ Sqrt@ mx}]][[2, 1]] (* Giovanni Resta, Mar 12 2017 *)
Select[Range[2*10^6],PrimeQ[DivisorSigma[1,#^3]]&] (* Harvey P. Dale, Jan 10 2024 *)

PARI

isok(n) = isprime(sigma(n^3)); \\ Michel Marcus, Mar 12 2017