A008334 Number of distinct primes dividing p-1, where p = n-th prime.

0, 1, 1, 2, 2, 2, 1, 2, 2, 2, 3, 2, 2, 3, 2, 2, 2, 3, 3, 3, 2, 3, 2, 2, 2, 2, 3, 2, 2, 2, 3, 3, 2, 3, 2, 3, 3, 2, 2, 2, 2, 3, 3, 2, 2, 3, 4, 3, 2, 3, 2, 3, 3, 2, 1, 2, 2, 3, 3, 3, 3, 2, 3, 3, 3, 2, 4, 3, 2, 3, 2, 2, 3, 3, 3, 2, 2, 3, 2, 3, 3, 4, 3, 2, 3, 3, 2, 3, 3, 4, 2, 2, 2, 3, 3

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn

Comments

This is omega(p-1), i.e. A001221(A006093(n)). For Omega(p-1) = A001222(A006093(n)), see A023508. - Lekraj Beedassy, Oct 08 2004

Primes counted without multiplicity. - Harvey P. Dale, May 05 2018

References

• N. P. Ryzhova, Asymptotic formulae in a binary problem of shifted prime numbers (in Russian), Additive problems of number theory, Interuniv. Collect. Sci. Works, Kujbyshev 1985 (1985), pp. 25-31.

Links

• T. D. Noe, Table of n, a(n) for n=1..10000
• H. Halberstam, On the Distribution of Additive Number-Theoretic Functions (III), Journal of the London Mathematical Society, Vol. s1-31, No. 1 (1956), pp. 14-27.
• C. B. Haselgrove, Some Theorems in the Analytic Theory of Numbers, Journal of the London Mathematical Society, Vol. s1-26, No. 4 (1951), pp. 273-277.

Crossrefs

Cf. A001221, A001222, A006093, A023508.

Programs

Maple

for i from 1 to 500 do if isprime(i) then print(nops(factorset(i-1))); fi; od;

Mathematica

PrimeNu[#]&/@(Prime[Range[100]]-1) (* Harvey P. Dale, May 05 2018 *)

PARI

a(n,p=prime(n))=omega(p-1) \\ Charles R Greathouse IV, Nov 29 2024

Formula

Sum_{k; prime(k)<=n} a(k) = n*log(log(n))/log(n) + O(n/log(n)) (Haselgrove, 1951; Halberstam, 1956; Ryzhova, 1985). - Amiram Eldar, Mar 05 2021 [corrected by Charles R Greathouse IV, Nov 29 2024]

Extensions

Definition clarified by Harvey P. Dale, May 05 2018