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

%I A008334 #25 Jul 08 2025 01:02:44
%S A008334 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,
%T A008334 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,
%U A008334 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
%N A008334 Number of distinct primes dividing p-1, where p = n-th prime.
%C A008334 This is omega(p-1), i.e. A001221(A006093(n)). For Omega(p-1) = A001222(A006093(n)), see A023508. - _Lekraj Beedassy_, Oct 08 2004
%C A008334 Primes counted without multiplicity. - _Harvey P. Dale_, May 05 2018
%D A008334 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.
%H A008334 T. D. Noe, <a href="/A008334/b008334.txt">Table of n, a(n) for n=1..10000</a>
%H A008334 H. Halberstam, <a href="https://doi.org/10.1112/jlms/s1-31.1.14">On the Distribution of Additive Number-Theoretic Functions (III)</a>, Journal of the London Mathematical Society, Vol. s1-31, No. 1 (1956), pp. 14-27.
%H A008334 C. B. Haselgrove, <a href="https://doi.org/10.1112/jlms/s1-26.4.273">Some Theorems in the Analytic Theory of Numbers</a>, Journal of the London Mathematical Society, Vol. s1-26, No. 4 (1951), pp. 273-277.
%F A008334 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]
%p A008334 for i from 1 to 500 do if isprime(i) then print(nops(factorset(i-1))); fi; od;
%t A008334 PrimeNu[#]&/@(Prime[Range[100]]-1) (* _Harvey P. Dale_, May 05 2018 *)
%o A008334 (PARI) a(n,p=prime(n))=omega(p-1) \\ _Charles R Greathouse IV_, Nov 29 2024
%Y A008334 Cf. A001221, A001222, A006093, A023508.
%K A008334 nonn
%O A008334 1,4
%A A008334 _N. J. A. Sloane_
%E A008334 Definition clarified by _Harvey P. Dale_, May 05 2018