A281882 Numbers n such that the number of divisors of sum of divisors of n is prime.

2, 3, 4, 9, 16, 25, 64, 81, 289, 400, 651, 729, 889, 1681, 2401, 2667, 3481, 3937, 4096, 5041, 7921, 10201, 15625, 17161, 27889, 28561, 29929, 57337, 65536, 83521, 85849, 146689, 172011, 253921, 262144, 279841, 458329, 491401, 531441, 552049, 579121, 597529

Offset

1,1

Comments

Numbers n such that A062068(n) = A000005(A000203(n)) = tau(sigma(n)) is a prime.

Conjecture: 2 and 3 are the only prime numbers in the sequence.

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

Links

Charles R Greathouse IV, Table of n, a(n) for n = 1..1500

Example

81 is a term because tau(sigma(81)) = tau(121) = 3 (prime).
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, ...

Maple

with(numtheory): A281882:=n->`if`(isprime(tau(sigma(n))), n, NULL): seq(A281882(n), n=1..10^5); # Wesley Ivan Hurt, Feb 01 2017

Mathematica

Select[Range[10^6], PrimeQ@ DivisorSigma[0, DivisorSigma[1, #]] &] (* Michael De Vlieger, Feb 04 2017 *)

Prog

(Magma) [n: n in[1..10^7] | IsPrime(NumberOfDivisors(SumOfDivisors(n)))]
(PARI) isok(n) = isprime(numdiv(sigma(n))); \\ Michel Marcus, Feb 01 2017

Crossrefs

Supersequence of A023194.

Cf. A000005, A000203, A009087, A062068.

Sequence in context: A098969 A235401 A354268 * A325436 A346777 A338585

Adjacent sequences: A281879 A281880 A281881 * A281883 A281884 A281885

Keyword

nonn

Author

Jaroslav Krizek, Feb 01 2017

Status

approved