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 A160642 Minimal number k such that n! can be written as product of k (>= 2) consecutive integers. 0
 2, 2, 3, 3, 3, 4, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 20, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72 (list; graph; refs; listen; history; text; internal format)
 OFFSET 2,1 COMMENTS Sequence starts at n=2 because 1! cannot be written as product of 2 (or more) consecutive integers. For suitable m >= n, we have n! = m!/(m-a(n))! For n >= 3, we have a(n) <= n-1 because n! = 2*...*n. For n = m! - 1, we have a(n) <= m!-m because n! = (m+1)*(m+2)*...*(m!-1)*m! = (n+1)!/m!. For n>=8, it appears that the preceding two inequalities completely describe a(n), i.e. a(n) = m!-m if n=m!-1 and a(n)=n-1 otherwise. LINKS EXAMPLE a(2) = 2 because 2! = 1*2. a(3) = 2 because 3! = 2*3. a(4) = 3 because 4! = 2*3*4. a(5) = 3 because 5! = 4*5*6. a(6) = 3 because 6! = 8*9*10. a(7) = 4 because 7! = 7*8*9*10. PROG (PARI) csfac(N, k) = local(d, w=floor(N^(1/k))); while((d=prod(i=1, k, w+i))>N, w=w-1); if(d==N, 1, 0) csmin(N) = local(k=2); while(csfac(N, k)==0, k=k+1); k \p 200; for(n=2, 200, print(csmin(n!))) CROSSREFS Sequence in context: A331590 A326496 A058740 * A110868 A110869 A110876 Adjacent sequences:  A160639 A160640 A160641 * A160643 A160644 A160645 KEYWORD nonn AUTHOR Hagen von Eitzen, May 21 2009 STATUS approved

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Last modified September 23 16:41 EDT 2020. Contains 337315 sequences. (Running on oeis4.)