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 A319404 a(n) is the period of the periodic k-sequence q_k=lcm(k+1,k+2,...,k+n)/(n*binomial(k+n,n)). 0
 1, 1, 2, 3, 12, 20, 60, 105, 280, 504, 2520, 27720, 27720, 51480, 72072, 45045, 720720, 1361360, 12252240, 46558512, 33256080, 21162960, 232792560, 5354228880, 1070845776, 2059318800, 2974571600, 11473347600, 80313433200, 2329089562800, 2329089562800, 4512611027925 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,3 COMMENTS For n>0, k>=0, the k-sequence q_k=lcm(k+1,k+2,...,k+n)/(n*binomial(k+n,n)) is a periodic integer sequence with period a(n). a(n) is a divisor of A003418(n-1) and a multiple of A003418(n)/n. a(n) = A003418(n-1) if n is a member of A027854 (a mutinous number), otherwise a(n) = A003418(n)/q^v where q^v is the highest prime power which divides n. a(n) = A003418(n-1) iff n is a mutinous number or n is a prime number. a(n) = A003418(n) iff n is a mutinous number. lcm(k+1,k+2,...,k+n)/(n*binomial(k+n,n)) is a divisor of lcm(1,2,...,n)/n, therefore a(n) is also the period of the periodic k-sequence r_k= binomial(k+n,n)*lcm(1,2,...,n)/lcm(k+1,k+2,...,k+n). Let g be the smallest multiple of A003418(n)/n such that r_g=r_0=1 and r_{g+1}=r_1=gcd(m+1,A003418(n)), then a(n)=g. a(n+j) is a multiple of binomial(n+j-1,j). All these statements require proofs. LINKS EXAMPLE For n = 5, a(5) = 12 since from k>=0, we have lcm(k+1,k+2,k+3,k+4,k+5)/5/binomial(k+5,5) =  12,2,4,3,4,2,12,1,4,6,4,1,12,2,4,3,4,2,12,1,4,6,4,1,12,2,4,3,4,2,12,1,4,6,4,1,12,..., etc. a periodic sequence of period 12. MATHEMATICA ll2[n0_, m0_] := Module[{f, g, i, n = n0, m = m0}, g = 1;   If[1 <= m <= n, Do[f = LCM[g, n - i]; g = f, {i, 0, m - 1}], f = 1]; f] list3 = {1}; Do[i = 0; ll = ll2[m, m]/m; b = {1, ll }; a = {0, 0 };   While[ a != b, i = i + ll;    a = { ll2[m + i - 1, m]/(m*Binomial[m + i - 1, m]), ll2[m + i, m]/(      m*Binomial[m + i, m])}]; AppendTo[list3, i], {m, 2, 50}]; Print[list3] CROSSREFS Cf. A003418, A027854. Sequence in context: A130089 A323119 A307035 * A126292 A083265 A067391 Adjacent sequences:  A319401 A319402 A319403 * A319405 A319406 A319407 KEYWORD nonn AUTHOR René Gy, Sep 18 2018 STATUS approved

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Last modified November 19 03:44 EST 2019. Contains 329310 sequences. (Running on oeis4.)