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

%I A299147 #36 Mar 02 2025 14:33:43
%S A299147 4,64,289,253541929,499477801,1260747049,14450203681,25391466409,
%T A299147 256221229489,333456586849,341122579249,459926756041,911087431081,
%U A299147 928731181849,1142288550841,2880002461249,2923070670601,3000305515321,4103999343889,4123226708329,4258977385441
%N A299147 Numbers k such that sigma(k), sigma(k^2) and sigma(k^3) are primes.
%C A299147 All terms are squares (proof in A023194).
%C A299147 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).
%H A299147 Chai Wah Wu, <a href="/A299147/b299147.txt">Table of n, a(n) for n = 1..12775</a> (n = 1..997 from Robert G. Wilson v)
%e A299147 4 is in the sequence because all sigma(4) = 7, sigma(4^2) = 31 and sigma(4^3) = 127 are primes.
%p A299147 N:= 10^14: # to get all terms <= N
%p A299147 Res:= NULL:
%p A299147 p:= 1:
%p A299147 do
%p A299147   p:= nextprime(p);
%p A299147   if p^2 > N then break fi;
%p A299147   for k from 2 by 2 while p^k <= N do
%p A299147     if isprime(k+1) and isprime(2*k+1) and isprime(3*k+1) then
%p A299147       q1:= (p^(k+1)-1)/(p-1);
%p A299147       q2:= (p^(2*k+1)-1)/(p-1);
%p A299147       q3:= (p^(3*k+1)-1)/(p-1);
%p A299147       if isprime(q1) and isprime(q2) and isprime(q3) then
%p A299147         Res:= Res, p^k;
%p A299147       fi
%p A299147     fi
%p A299147   od
%p A299147 od:
%p A299147 sort([Res]); # _Robert Israel_, Feb 22 2018
%t A299147 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 *)
%o A299147 (Magma) [n: n in[1..10000000] | IsPrime(SumOfDivisors(n)) and IsPrime(SumOfDivisors(n^2)) and IsPrime(SumOfDivisors(n^3))];
%o A299147 (PARI) isok(n) = isprime(sigma(n)) && isprime(sigma(n^2)) && isprime(sigma(n^3)); \\ _Michel Marcus_, Feb 05 2018
%Y A299147 Subsequence of A232444.
%Y A299147 Cf. A000203, A055638, A279094, A279096, A299153.
%K A299147 nonn
%O A299147 1,1
%A A299147 _Jaroslav Krizek_, Feb 03 2018