A023503 Greatest prime divisor of prime(n) - 1.

2, 2, 3, 5, 3, 2, 3, 11, 7, 5, 3, 5, 7, 23, 13, 29, 5, 11, 7, 3, 13, 41, 11, 3, 5, 17, 53, 3, 7, 7, 13, 17, 23, 37, 5, 13, 3, 83, 43, 89, 5, 19, 3, 7, 11, 7, 37, 113, 19, 29, 17, 5, 5, 2, 131, 67, 5, 23, 7, 47, 73, 17, 31, 13, 79, 11, 7, 173, 29, 11, 179, 61, 31, 7, 191

Offset: 2

Author

Clark Kimberling

Keywords

• nonn

Comments

Baker & Harman (1998) show that there are infinitely many n such that a(n) > prime(n)^0.677. This improves on earlier work of Goldfeld, Hooley, Fouvry, Deshouillers, Iwaniec, Motohashi, et al.

Fouvry shows that a(n) > prime(n)^0.6683 for a positive proportion of members of this sequence. See Fouvry and also Baker & Harman (1996) which corrected an error in the former work.

The record values are the Sophie Germain primes A005384. - Daniel Suteu, May 09 2017

Conjecture: every prime is in the sequence. Cf. A035095 (see my comment). - Thomas Ordowski, Aug 06 2017

a(n) is 2 for n in A159611, and is at most 3 for n in A174099. Conjecture: liminf a(n) = 3. - Jeppe Stig Nielsen, Jul 04 2020

Links

• T. D. Noe, Table of n, a(n) for n = 2..10000
• R. C. Baker and G. Harman, The Brun-Titchmarsh theorem on average, Analytic Number Theory (Proceedings in honor of Heini Halberstam), Birkhäuser, Boston, 1996, pp. 39-103.
• R. Baker and G. Harman, Shifted primes without large prime factors, Acta Arithmetica 83 (1998), pp. 331-361.
• Étienne Fouvry, Théorème de Brun-Titchmarsh; application au théorème de Fermat, Invent. Math 79 (1985), 383-407.
• D. M. Goldfeld, On the number of primes p for which p + a has a large prime factor, Mathematika 16 (1969), pp. 23-27.
• R. R. Hall, Some properties of the sequence {p-1}, Acta Arith. 28 (1975/76), 101-105.
• G. Harman, On the greatest prime factor of p-1 with effective constants

Crossrefs

Cf. A006093, A006530.

Cf. A005384, A035095, A159611, A174099.

Programs

Maple

A023503 := proc(n)
    A006530(ithprime(n)-1) ;
end proc:
seq( A023503(n),n=2..80) ; # R. J. Mathar, Sep 07 2016

Mathematica

Table[FactorInteger[Prime[n] - 1][[-1, 1]], {n, 2, 100}] (* T. D. Noe, Jun 08 2011 *)

PARI

a(n) = vecmax(factor(prime(n)-1)[,1]); \\ Michel Marcus, Aug 15 2015

Formula

a(n) = A006530(A006093(n)). - Michel Marcus, Aug 15 2015

Extensions

Comments, references, and links from Charles R Greathouse IV, Mar 04 2011