A278741 Odd numbers k such that tau(k-1) is a prime.

3, 5, 17, 65, 1025, 4097, 65537, 262145, 4194305, 268435457, 1073741825, 68719476737, 1099511627777, 4398046511105, 70368744177665, 4503599627370497, 288230376151711745, 1152921504606846977, 73786976294838206465, 1180591620717411303425, 4722366482869645213697

Offset: 1

Author

Jaroslav Krizek, Nov 27 2016

Keywords

• nonn

Comments

tau(k) = A000005(k) = the number of divisors of k.

Conjecture: prime terms are in A249759: 3, 5, 17, 65537, ...

Supersequence of A256438 and A249759. Subsequence of {A009087(n) + 1}.

Examples

			Odd number 65 is in the sequence because tau(64) = 7 (prime).
		

Crossrefs

Cf. A000005, A000040, A001348, A009087, A061286, A249759, A256438.

Programs

Magma

[n: n in[2..10000000] |  IsOdd(n) and IsPrime(NumberOfDivisors(n-1))];

PARI

isok(n) = (n % 2) && isprime(numdiv(n-1)); \\ Michel Marcus, Nov 27 2016

Formula

a(n) = A061286(n) + 1.

sigma(a(n)-1) = A001348(n), i.e., Mersenne numbers.

tau(a(n)-1) = A000040(n), i.e., all primes; a(n) = the smallest odd number k such that tau(a(n)-1) = prime(n) = A000040(n).