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A063980
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Pillai primes: primes p such that there exists an integer m such that m! + 1 == 0 (mod p) and p != 1 (mod m).
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6
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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
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OFFSET
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1,1
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COMMENTS
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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
Named after the Indian mathematician Subbayya Sivasankaranarayana Pillai (1901-1950). - Amiram Eldar, Jun 16 2021
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LINKS
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MATHEMATICA
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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-François 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 *)
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PROG
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(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
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CROSSREFS
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KEYWORD
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nonn,nice
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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