A305398 Index of the least prime not dividing p-1, where p = A073918(n) is the smallest prime such that p-1 has n distinct prime factors.

1, 2, 3, 4, 5, 6, 6, 7, 7, 9, 10, 12, 11, 12, 13, 15, 16, 15, 16, 18, 19, 20, 21, 22, 22, 23, 25, 27, 26, 29, 29, 27, 31, 33, 32, 34, 36, 36, 38, 39, 35, 38, 40, 43, 38, 44, 46, 46, 45, 48, 50, 49, 49, 51, 50, 54, 54, 57, 58, 56, 57, 58, 58, 63, 62, 64, 63, 67, 64, 68, 69, 69, 70, 69, 74, 76, 71, 73, 76, 78, 80, 79, 80, 81, 84, 84, 83, 87, 88, 86, 88

Offset: 0

Author

M. F. Hasler, May 31 2018

Keywords

• nonn

Comments

For 0 <= n <= 5, A073918(n) = A002110(n) + 1 = prime(n)# + 1, therefore a(n) = n + 1. From n >= 6 on, some smaller primes are missing in the factorization of A073918(n) - 1, whence a(n) <= n.

This is related to the conjecture formulated in A073918, that for any m there is K(m) such that prime(m)# | A073918(k)-1 for all k >= K(m): This conjecture is equivalent to lim inf a(n) = oo.

Examples

			For 0 <= n <= 5, the smallest prime p = A073918(n) such that p-1 has n distinct prime factors is p = prime(n)# + 1, therefore a(n) = n + 1 is the index of the smallest prime not dividing p - 1.
For n = 6, the smallest prime p such that p - 1 has 6 distinct prime factors is prime(5)#*prime(8) + 1, therefore a(n) = 6.
		

Crossrefs

Cf. A073918, A002110.

Programs

PARI

a(n)={(n=factor(A073918(n)-1)[,1])&& for(i=2,#n,n[i]>prime(i)&&return(i)); #n+1} \\ For illustration; it is more efficient to adapt code from A073918 to compute the sequence.