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
KEYWORD
nonn
AUTHOR
Jaroslav Krizek, Feb 03 2018
STATUS
approved