Search: id:A059751
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%I A059751
%S A059751 7,6,5,10,9,14,13,12,15,22,21,20,19,20,23,22,21,20,19,18,27,26,25,24,29,
%T A059751 30,29,28,27,26,25,24,31,34,41,40,39,46,47,46,45,44,43,42,41,44,43,42,
%U A059751 43,42,43,42,41,40,55,54,53,60,59,58,57,58,57,56,57,56,55,54,59,58,57
%N A059751 Grimm numbers (2): a(n) = largest k so that for each composite m in {n+1, 
               n+2, ..., n+k} there corresponds a different divisor d_m with 1 < 
               d_m < m.
%C A059751 Comment from T. D. Noe, Feb 18 2009: Erdos and Pomerance conjectured 
               that the number n+a(n)+1, which "blocks" a(n) from becoming larger, 
               is always an odd semiprime. They verified this conjecture up to n=492 
               and proved it for large n. The numbers n at which n+a(n)+1 increases 
               also appear to be semiprimes.
%D A059751 C. A. Grimm, A conjecture on consecutive composite numbers, Amer. Math. 
               Monthly, 76 (1969), 1126-1128.
%D A059751 D. S. Mitrinovic et al., Handbook of Number Theory, Kluwer, Section XII.15, 
               p. 438.
%D A059751 Paul Erdos and Carl Pomerance, An analogue of Grimm's problem of finding 
               distinct prime factors of consecutive integers, Util. Math. 24 (1983), 
               45-65. [From T. D. Noe (noe(AT)sspectra.com), Feb 17 2009]
%H A059751 T. D. Noe, <a href="b059751.txt">Table of n, a(n) for n=1..1000</a>
%e A059751 For n=4 we look at the sequence {5, 6, 7, ... } and we must choose distinct 
               proper divisors for as many composites as we can. We can choose 2 
               for 6, 4 for 8, 3 for 9, 5 for 10, 6 for 12 and 7 for 14, but now 
               all the proper divisors of 15 have appeared, so we stop and a(4) 
               = 14-4 = 10.
%Y A059751 Cf. A059686, A059752.
%Y A059751 Sequence in context: A104178 A092874 A015791 this_sequence A019859 A102769 
               A031348
%Y A059751 Adjacent sequences: A059748 A059749 A059750 this_sequence A059752 A059753 
               A059754
%K A059751 nonn,easy,nice
%O A059751 1,1
%A A059751 N. J. A. Sloane (njas(AT)research.att.com), Feb 11 2001
%E A059751 More terms from Naohiro Nomoto (6284968128(AT)geocities.co.jp), Mar 03 
               2001
%E A059751 Extended by T. D. Noe (noe(AT)sspectra.com), Feb 17 2009

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