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A230459 Ordered by increasing m with k<m, a(n) is the n-th record value of gcd(k!+1,m!+1). 1
2, 7, 71, 661, 733, 2371, 3529, 13499, 46549, 98101, 163517, 197933, 1924217, 3322441, 5370731 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

The pairs (m,k) generating records are (1,0), (6,3), (9,7), (17,8), (89,51), (174,144), (349,228), (422,81), (650,406), (1415,1718), (1697,161), (1622,773), (1884,1219), (7003,2031) and (17057,660).

Heuristics in concert with a database of 'small' (less than, say, 10^12) prime factors of numbers of this kind would generate faster accurate results with near certainty, while any truly proving program is doomed to be relatively slow by comparison (and see following on a(15)).

Note: An auxiliary program employed a limit of 10^8--in lieu of a database--to generate the likely-but-not-certain value of a(15) shown last.

LINKS

Table of n, a(n) for n=1..15.

EXAMPLE

a(1)=2, corresponding to m=1 and k=0.  7 is the first value other than 1 to be the greatest common divisor of two different numbers k!+1 and m!+1, where m is increasing and k is allowed to increase to m-1 for a given m (For m=6 and k=3, m!+1=7*103 and k!+1=7); so that a(2)=7.

PROG

(PARI)

{

\\ The constant L here is arbitrary.\\

\\ This does not generate a(1).\\

rec=2; L=10000; F=vector(L); n=2;

for(k=1, L, n--; n*=k; n++; F[k]=n);

for(m=2, L,

  for(k=1, m-1,

    a=gcd(F[m], F[k]); if(a>rec,

      rec=a; print1(a": "m", "k"\n"))))

}

CROSSREFS

A038507, A051301, A002583, A002981

Sequence in context: A106917 A188665 A198188 * A267406 A100360 A061421

Adjacent sequences:  A230456 A230457 A230458 * A230460 A230461 A230462

KEYWORD

nonn,more

AUTHOR

James G. Merickel, Oct 19 2013

STATUS

approved

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Last modified July 28 12:39 EDT 2017. Contains 289889 sequences.