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 A230459 Ordered by increasing m with k
 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 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 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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