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A023503
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Greatest prime divisor of prime(n) - 1.
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16
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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
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OFFSET
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2,1
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COMMENTS
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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
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LINKS
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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), Birkhauser, Boston, 1996, pp. 39-103.
R. Baker and G. Harman, Shifted primes without large prime factors, Acta Arithmetica 83 (1998), pp. 331-361.
Etienne 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
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FORMULA
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a(n) = A006530(A006093(n)). - Michel Marcus, Aug 15 2015
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MAPLE
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A023503 := proc(n)
A006530(ithprime(n)-1) ;
end proc:
seq( A023503(n), n=2..80) ; # R. J. Mathar, Sep 07 2016
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MATHEMATICA
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Table[FactorInteger[Prime[n] - 1][[-1, 1]], {n, 2, 100}] (* T. D. Noe, Jun 08 2011 *)
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PROG
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(PARI) a(n) = vecmax(factor(prime(n)-1)[, 1]); \\ Michel Marcus, Aug 15 2015
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CROSSREFS
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Cf. A006093, A006530.
Sequence in context: A203955 A039638 A090926 * A241195 A039640 A053811
Adjacent sequences: A023500 A023501 A023502 * A023504 A023505 A023506
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KEYWORD
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nonn
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AUTHOR
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Clark Kimberling
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EXTENSIONS
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Comments, references, and links from Charles R Greathouse IV, Mar 04 2011
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STATUS
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approved
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