A100366 a(n) is the least prime number q such that q,q+1,q+2,q+3,...,q+n-1 have 2,4,8,...,2^n divisors respectively.

2, 5, 193, 613, 1124581, 52071301, 213536830501

Offset: 1

Author

Labos Elemer, Nov 19 2004

Keywords

• nonn
• fini
• full

Comments

a(3), a(4), a(5) are the initial terms of A100363, A100364, A100365 resp.

Any run of 8 or more consecutive integers must include at least one number k of the form 8j+4; in the prime factorization of k, the prime factor 2 appears with multiplicity exactly 2, so the number of divisors of k is divisible by 3 (which is not a power of 2). Thus, there is no term a(8): the sequence is complete, ending with a(7). - Jon E. Schoenfield, Nov 12 2017

Examples

			a(4)=613: q=613 (a prime, hence two divisors), q+1 = 614 = 2*307 (4 divisors), q+2 = 615 = 3*5*41 (8 divisors), and q+3 = 616 = 2^3 * 7 * 11 (16 divisors).
		

Crossrefs

Cf. A000005, A063446, A100363, A100364.

Extensions

a(6)-a(7) from Donovan Johnson, Mar 23 2011

Keywords fini and full added and Example section edited by Jon E. Schoenfield, Nov 12 2017