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A283262 Numbers m such that tau(m^2) is a prime.

%I A283262 #24 Mar 17 2025 16:35:53
%S A283262 2,3,4,5,7,8,9,11,13,17,19,23,25,27,29,31,32,37,41,43,47,49,53,59,61,
%T A283262 64,67,71,73,79,83,89,97,101,103,107,109,113,121,125,127,131,137,139,
%U A283262 149,151,157,163,167,169,173,179,181,191,193,197,199,211,223,227
%N A283262 Numbers m such that tau(m^2) is a prime.
%C A283262 tau(m) is the number of positive divisors of m (A000005).
%C A283262 Numbers m such that A000005(A000290(m)) = A048691(m) is a prime.
%C A283262 Union of A000040 (primes) and A051676.
%C A283262 Supersequence of A055638 (sigma(m^2) is prime).
%C A283262 Subsequence of A000961 (powers of primes).
%C A283262 Prime powers p^e with 2e+1 prime (e >= 1).
%C A283262 See A061285(m) = the smallest number k such that tau(k^2) = m-th prime.
%H A283262 Robert Israel, <a href="/A283262/b283262.txt">Table of n, a(n) for n = 1..10000</a>
%e A283262 tau(4^2) = tau(16) = 5 (prime).
%p A283262 N:= 1000: # to get all terms <= N
%p A283262 es:= select(t -> isprime(2*t+1), [$1..ilog2(N)]):
%p A283262 Ps:= select(isprime, [2,seq(i,i=3..N,2)]):
%p A283262 sort(select(`<=`, [seq(seq(p^e,e=es),p=Ps)],N)): # _Robert Israel_, Mar 16 2017
%t A283262 Select[Range@ 227, PrimeQ[DivisorSigma[0, #^2]] &] (* _Michael De Vlieger_, Mar 09 2017 *)
%o A283262 (Magma) [n: n in [2..100000] | IsPrime(NumberOfDivisors(n^2))];
%o A283262 (PARI) isok(n)=isprime(numdiv(n^2)) \\ _Indranil Ghosh_, Mar 09 2017
%Y A283262 Cf. A000005, A000290, A000961, A048691, A051676, A055638, A061285.
%K A283262 nonn
%O A283262 1,1
%A A283262 _Jaroslav Krizek_, Mar 08 2017