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

0,2

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.

Links

Table of n, a(n) for n=0..90.

Example

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.

Prog

(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.

Crossrefs

Cf. A073918, A002110.

Sequence in context: A005376 A196362 A195879 * A216522 A086419 A287354

Adjacent sequences: A305395 A305396 A305397 * A305399 A305400 A305401

Keyword

nonn

Author

M. F. Hasler, May 31 2018

Status

approved