A299148 a(n) is the smallest number k such that sigma(k) and sigma(k^n) are both primes.

2, 2, 4, 2, 25, 2, 262144, 4, 4, 64, 734449, 2, 3100870943041, 9066121, 4, 2, 729, 2, 214355670008317962105386619478205641151753401, 5041, 64, 16, 25, 10651330026288961, 16610312161, 2607021481, 38950081, 1817762776525603445521, 5331481, 2, 2160067977820518171249529658520145004718584607049, 21203610154988994565561

Offset

1,1

Comments

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

a(n) is of the form p^e where p, e+1 and e*n+1 are primes. e=1 is possible only in the case p=2. - Robert Israel, Feb 06 2018

Links

Robert Israel, Table of n, a(n) for n = 1..79

Formula

a(n) >= A279094(n).

Example

For n = 3; a(3) = 4 because 4 is the smallest number such that sigma(4) = 7 and sigma(4^3) = 127 are both primes.

Maple

f:= proc(n, Nmin, Nmax) local p, e, M, Res;
  M:= Nmax;
  Res:= -1;
  e:= 0;
  do
    e:= nextprime(e+1)-1;
    if 2^e > M then return Res fi;
    if not isprime(e*n+1) then next fi;
    p:= floor(Nmin^(1/e));
    do
      p:= nextprime(p);
      if p^e > M then break fi;
      if e = 1 and p > 2 then break fi;
      if isprime((p^(e+1)-1)/(p-1)) and isprime((p^(e*n+1)-1)/(p-1)) then
        Res:= p^e;
        M:= p^e;
        break
      fi
    od
  od;
end proc:
g:= proc(n) local Nmin, Nmax, v;
  Nmax:= 1;
  do
    Nmin:= Nmax;
    Nmax:= Nmax*10^3;
    v:= f(n, Nmin, Nmax);
    if v > 0 then return v fi;
  od;
end proc:
seq(g(n), n=1..50); # Robert Israel, Feb 06 2018

Mathematica

Array[Block[{k = 2}, While[! AllTrue[DivisorSigma[1, #] & /@ {k, k^#}, PrimeQ], k++]; k] &, 10] (* Michael De Vlieger, Feb 05 2018 *)

Prog

(Magma) [Min([n: n in[1..10000000] | IsPrime(SumOfDivisors(n)) and IsPrime(SumOfDivisors(n^k))]): k in [2..12]]
(PARI) a(n) = {my(k=1); while (!(isprime(sigma(k)) && isprime(sigma(k^n))), k++); k; } \\ Michel Marcus, Feb 05 2018

Crossrefs

Cf. A000203, A279094.

Sequence in context: A333595 A227509 A279094 * A129243 A013551 A252040

Adjacent sequences: A299145 A299146 A299147 * A299149 A299150 A299151

Keyword

nonn

Author

Jaroslav Krizek, Feb 03 2018

Extensions

a(13) to a(32) from Robert Israel, Feb 06 2018

Status

approved