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%I #24 Sep 08 2022 08:46:18
%S 2,3,4,9,16,25,64,81,289,400,651,729,889,1681,2401,2667,3481,3937,
%T 4096,5041,7921,10201,15625,17161,27889,28561,29929,57337,65536,83521,
%U 85849,146689,172011,253921,262144,279841,458329,491401,531441,552049,579121,597529
%N Numbers n such that the number of divisors of sum of divisors of n is prime.
%C Numbers n such that A062068(n) = A000005(A000203(n)) = tau(sigma(n)) is a prime.
%C Conjecture: 2 and 3 are the only prime numbers in the sequence.
%C Proof: If p is an odd prime, then sigma(p) = p+1 is even and hence the number of divisors of sigma(p) is composite unless p+1 is a power of 2, in which case let p+1 = 2^k which has k+1 divisors. But then p = 2^k - 1 is divisible by 2^(k/2) - 1, which is greater than 1 for p > 3. - _Charles R Greathouse IV_, Feb 01 2017
%H Charles R Greathouse IV, <a href="/A281882/b281882.txt">Table of n, a(n) for n = 1..1500</a>
%e 81 is a term because tau(sigma(81)) = tau(121) = 3 (prime).
%e For n>=1; tau(a(n)) = 2, 3, 2, 2, 2, 2, 2, 3, 2, 3, 11, 2, 11, 2, 2, 13, 2, 13, 2, 2, 2, 2, 2, 2, 2, 2, 2, 17, 2, 2, 2, ...
%p with(numtheory): A281882:=n->`if`(isprime(tau(sigma(n))), n, NULL): seq(A281882(n), n=1..10^5); # _Wesley Ivan Hurt_, Feb 01 2017
%t Select[Range[10^6], PrimeQ@ DivisorSigma[0, DivisorSigma[1, #]] &] (* _Michael De Vlieger_, Feb 04 2017 *)
%o (Magma) [n: n in[1..10^7] | IsPrime(NumberOfDivisors(SumOfDivisors(n)))]
%o (PARI) isok(n) = isprime(numdiv(sigma(n))); \\ _Michel Marcus_, Feb 01 2017
%Y Supersequence of A023194.
%Y Cf. A000005, A000203, A009087, A062068.
%K nonn
%O 1,1
%A _Jaroslav Krizek_, Feb 01 2017
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