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%I #18 Jan 10 2024 12:37:45
%S 4,9,16,25,64,81,169,289,625,961,1024,2401,3721,5329,7921,22201,26569,
%T 63001,121801,124609,212521,273529,358801,418609,744769,885481,896809,
%U 1048576,1181569,1247689,1510441,1630729,1666681,1682209,1771561,1874161,1985281
%N Numbers k such that sigma(k^3) is prime.
%C All terms are square. Moreover, each term is of the form p^j where both p and j*3 + 1 are prime (see A279094).
%H Giovanni Resta, <a href="/A279096/b279096.txt">Table of n, a(n) for n = 1..10000</a>
%e 4 is in the sequence because sigma(4^3) = sigma(2^6) = 1 + 2 + 4 + 8 + 16 + 32 + 64 = 127, which is prime.
%e 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.
%e 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.)
%e 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.
%t 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 *)
%t Select[Range[2*10^6],PrimeQ[DivisorSigma[1,#^3]]&] (* _Harvey P. Dale_, Jan 10 2024 *)
%o (PARI) isok(n) = isprime(sigma(n^3)); \\ _Michel Marcus_, Mar 12 2017
%Y 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).
%K nonn
%O 1,1
%A _Jon E. Schoenfield_, Mar 12 2017
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