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A063980 Pillai primes: p such that there exists an integer m such that m!+1 is 0 mod p and p is not 1 mod m. 6
23, 29, 59, 61, 67, 71, 79, 83, 109, 137, 139, 149, 193, 227, 233, 239, 251, 257, 269, 271, 277, 293, 307, 311, 317, 359, 379, 383, 389, 397, 401, 419, 431, 449, 461, 463, 467, 479, 499, 503, 521, 557, 563, 569, 571, 577, 593, 599, 601, 607 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

Hardy & Subbarao prove that this sequence is infinite. An upper bound can be extracted from their proof: a(n) < e^e^...^e^O(n log n) with e appearing n times. This tetrational bound could be improved with results on the disjointness of the factorizations of numbers of the form k! + 1. - Charles R Greathouse IV, Sep 15 2015

LINKS

T. D. Noe and Charles R Greathouse IV, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe)

G. E. Hardy and M. V. Subbarao, A modified problem of Pillai and some related questions, Amer. Math. Monthly 109:6 (2002), pp. 554-559.

MATHEMATICA

ok[p_] := (r = False; Do[If[Mod[m! + 1, p] == 0 && Mod[p, m] != 1, r = True; Break[]], {m, 2, p}]; r); Select[Prime /@ Range[111], ok] (* Jean-Fran├žois Alcover, Apr 22 2011 *)

nn=1000; fact=1+Rest[FoldList[Times, 1, Range[nn]]]; t={}; Do[p=Prime[i]; m=2; While[m<p && !(Mod[p, m]!=1 && Mod[fact[[m]], p]==0), m++]; If[m<p, AppendTo[t, p]], {i, 2, PrimePi[nn]}]; t (* T. D. Noe, Apr 22 2011 *)

PROG

(PARI) is(p)=my(t=Mod(5040, p)); for(m=8, p-2, t*=m; if(t==-1 && p%m!=1, return(isprime(p)))); 0 \\ Charles R Greathouse IV, Feb 10 2013

CROSSREFS

Smallest m is given in A063828, largest in A211411.

Sequence in context: A045120 A174260 A097436 * A046124 A157677 A092518

Adjacent sequences:  A063977 A063978 A063979 * A063981 A063982 A063983

KEYWORD

nonn,nice

AUTHOR

R. K. Guy, Sep 08 2001

EXTENSIONS

More terms from David W. Wilson, Sep 08 2001

STATUS

approved

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Last modified December 9 06:57 EST 2016. Contains 278963 sequences.