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%I #37 Mar 28 2021 18:38:04
%S 1680,4320,120960,166320,175560,215760,725760,1080647568000
%N Numbers that can be written as a product of k consecutive composite numbers and also of k+1 consecutive composite numbers, for some k>1, with no factor used twice.
%C The term 725760 has three representations of 4, 5 and 6 numbers (with overlap).
%C From _David A. Corneth_, Mar 27 2021, Mar 28 2021: (Start)
%C a(12) > 10^22 if it exists.
%C Let x be the product of k consecutive composite numbers and y be the product of k+1 consecutive composite numbers giving some a(m). This sequence does not allow factors of x and y to overlap. If we do allow such overlaps we get A342876. (End)
%F Numbers of the form Product_{i=x..x+k} A002808(i) = Product_{i=y..y+k-1} A002808(i), where y > x + k and k > 1.
%e 1680 = 10*12*14 = 40*42.
%e 4320 = 6*8*9*10 = 15*16*18.
%e 120960 = 8*9*10*12*14 = 16*18*20*21.
%e 166320 = 18*20*21*22 = 54*55*56.
%e 175560 = 55*56*57 = 418*420.
%e 215760 = 58*60*62 = 464*465.
%e 725760 = 12*14*15*16*18 = 27*28*30*32.
%e 1080647568000 = 49*50*51*52*54*55*56 = 98*99*100*102*104*105.
%e From _David A. Corneth_, Mar 28 2021: (Start)
%e 1814400 = 8*9*10*12*14*15 = 15*16*18*20*21 is not in the sequence as the factor 15 is used more than once.
%e 104613949440000 = 12*14*15*16*18*20*21*22*24*25*26 = 20*21*22*24*25*26*27*28*30*32 is not here because, among others, the factor 20 is used more than once.
%e 115880067072000 = 4*6*8*9*10*12*14*15*16*18*20*21*22 = 8*9*10*12*14*15*16*18*20*21*22*24 is not here because most factors are used more than once. (End)
%Y Cf. A002808, A064224, A163263, A342876.
%K nonn,more
%O 1,1
%A _Manuel Valdivia_, Apr 16 2010, Apr 18 2010
%E Definition edited by _N. J. A. Sloane_, Apr 18 2010
%E Keyword:base removed by _R. J. Mathar_, Apr 24 2010
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