A299147 Numbers k such that sigma(k), sigma(k^2) and sigma(k^3) are primes.

4, 64, 289, 253541929, 499477801, 1260747049, 14450203681, 25391466409, 256221229489, 333456586849, 341122579249, 459926756041, 911087431081, 928731181849, 1142288550841, 2880002461249, 2923070670601, 3000305515321, 4103999343889, 4123226708329, 4258977385441

Offset

1,1

Comments

All terms are squares (proof in A023194).

Sequence {b(n)} of the smallest numbers m such that sigma(m^k) are primes for all k = 1..n: 2, 2, 4, ... (if fourth term exists, it must be greater than 10^16).

Links

Chai Wah Wu, Table of n, a(n) for n = 1..12775 (n = 1..997 from Robert G. Wilson v)

Example

4 is in the sequence because all sigma(4) = 7, sigma(4^2) = 31 and sigma(4^3) = 127 are primes.

Maple

N:= 10^14: # to get all terms <= N
Res:= NULL:
p:= 1:
do
  p:= nextprime(p);
  if p^2 > N then break fi;
  for k from 2 by 2 while p^k <= N do
    if isprime(k+1) and isprime(2*k+1) and isprime(3*k+1) then
      q1:= (p^(k+1)-1)/(p-1);
      q2:= (p^(2*k+1)-1)/(p-1);
      q3:= (p^(3*k+1)-1)/(p-1);
      if isprime(q1) and isprime(q2) and isprime(q3) then
        Res:= Res, p^k;
      fi
    fi
  od
od:
sort([Res]); # Robert Israel, Feb 22 2018

Mathematica

k = 1; A299147 = {}; While[k < 4260000000000, If[Union@ PrimeQ@ DivisorSigma[1, {k, k^2, k^3}] == {True}, AppendTo[A299147, k]]; k++]; A299147 (* Robert G. Wilson v, Feb 10 2018 *)

Prog

(Magma) [n: n in[1..10000000] | IsPrime(SumOfDivisors(n)) and IsPrime(SumOfDivisors(n^2)) and IsPrime(SumOfDivisors(n^3))]
(PARI) isok(n) = isprime(sigma(n)) && isprime(sigma(n^2)) && isprime(sigma(n^3)); \\ Michel Marcus, Feb 05 2018

Crossrefs

Subsequence of A232444.

Cf. A000203, A055638, A279094, A279096, A299153.

Sequence in context: A110258 A056982 A030994 * A141046 A264055 A222557

Adjacent sequences: A299144 A299145 A299146 * A299148 A299149 A299150

Keyword

nonn

Author

Jaroslav Krizek, Feb 03 2018

Status

approved