A299147 - OEIS

%I #32 Jan 21 2022 05:06:49

%S 4,64,289,253541929,499477801,1260747049,14450203681,25391466409,

%T 256221229489,333456586849,341122579249,459926756041,911087431081,

%U 928731181849,1142288550841,2880002461249,2923070670601,3000305515321,4103999343889,4123226708329,4258977385441

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

%C All terms are squares (proof in A023194).

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

%p N:= 10^14: # to get all terms <= N

%p Res:= NULL:

%p p:= 1:

%p do

%p   p:= nextprime(p);

%p   if p^2 > N then break fi;

%p   for k from 2 by 2 while p^k <= N do

%p     if isprime(k+1) and isprime(2*k+1) and isprime(3*k+1) then

%p       q1:= (p^(k+1)-1)/(p-1);

%p       q2:= (p^(2*k+1)-1)/(p-1);

%p       q3:= (p^(3*k+1)-1)/(p-1);

%p       if isprime(q1) and isprime(q2) and isprime(q3) then

%p         Res:= Res, p^k;

%p       fi

%p     fi

%p   od

%p od:

%p sort([Res]); # _Robert Israel_, Feb 22 2018

%t 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 (Magma) [n: n in[1..10000000] | IsPrime(SumOfDivisors(n)) and IsPrime(SumOfDivisors(n^2)) and IsPrime(SumOfDivisors(n^3))]

%o (PARI) isok(n) = isprime(sigma(n)) && isprime(sigma(n^2)) && isprime(sigma(n^3)); \\ _Michel Marcus_, Feb 05 2018

%Y Subsequence of A232444.

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

%K nonn

%O 1,1

%A _Jaroslav Krizek_, Feb 03 2018