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

Author

Jaroslav Krizek, Feb 03 2018

Keywords

• nonn

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

Examples

			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.
		

Links

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

Crossrefs

Cf. A000203, A279094.

Programs

Magma

[Min([n: n in[1..10000000] | IsPrime(SumOfDivisors(n)) and IsPrime(SumOfDivisors(n^k))]): k in [2..12]];

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 *)

PARI

a(n) = {my(k=1); while (!(isprime(sigma(k)) && isprime(sigma(k^n))), k++); k;} \\ Michel Marcus, Feb 05 2018

Formula

a(n) >= A279094(n).

Extensions

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