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A244656 Least product of consecutive positive integers which is divisible by each of 1, 2, ..., n.

%I A244656 #14 Mar 13 2018 04:08:23
%S A244656 2,2,6,12,60,60,420,840,2520,2520,55440,55440,360360,360360,360360,
%T A244656 2162160,85765680,85765680,33522128640,33522128640,33522128640,
%U A244656 33522128640,19275223968000,19275223968000,19275223968000
%N A244656 Least product of consecutive positive integers which is divisible by each of 1, 2, ..., n.
%C A244656 For n > 1, clearly a(n) is bounded below by lcm(1,2,...,n) and bounded above by n!. Further, a(n) is a positive multiple of lcm(1,2,...,n). Any product of two or more consecutive positive integers may be expressed as m!/k!, where 0 <= k <= m-2. For this sequence, the m corresponding to a(n) may or may not be a multiple of n. Whenever a(n) can be expressed as the product of exactly two consecutive integers, it is a term of A002378. See the a-file link for further comments.
%H A244656 Rick L. Shepherd, <a href="/A244656/b244656.txt">Table of n, a(n) for n = 1..36</a>
%H A244656 Rick L. Shepherd, <a href="/A244656/a244656.txt">Sample program output and calculation notes</a>
%e A244656 a(7) = 20*21 = 21!/19! = 420 because 420 is divisible by 1, 2, 3, 4, 5, 6, and 7, and no positive integer less than 420 is divisible by each of these. Here, 420 = lcm(1,2,3,4,5,6,7). 420 is an oblong (or promic) number (A002378).
%e A244656 a(11) = 7*8*9*10*11 = 11!/6! = 55440. Here, 27720 = lcm(1,2,3,4,5,6,7,8,9,10,11), but 27720 cannot be represented as a product of consecutive positive integers.
%e A244656 a(31) = 6081487775*6081487776 = 36984493563555938400, also a promic number.
%o A244656 (PARI)
%o A244656 {a(n) =
%o A244656 local(L, M, i, k = 0, s = 0, ret = 0, d, divs2,
%o A244656    st, pr, prt = 1); /* Use prt = 0 to suppress printing. */
%o A244656 if(n < 1, return, if(n < 3, ret = 2,
%o A244656 L = lcm(vector(n, i, i));
%o A244656 M = n!/L;
%o A244656 while(k < M,
%o A244656    k++;
%o A244656    s += L; d = divisors(s); divs2 = #d \ 2;
%o A244656    st = 2; pr = d[st];
%o A244656    i = 0;
%o A244656    while(st + i <= divs2,
%o A244656       if(d[st + i + 1] == d[st + i] + 1,
%o A244656          pr *= d[st + i + 1]; i++;
%o A244656          if(pr == s,
%o A244656             if(prt,
%o A244656                print1("k*L = ", k, "*", L, " = ",
%o A244656                  s, " = ", d[st], "*");
%o A244656                if(d[st + i] > d[st] + 2, print1("...*"),
%o A244656                  if(d[st + i] == d[st] + 2,
%o A244656                    print1(d[st] + 1, "*")));
%o A244656                print(d[st + i], " = ", d[st + i], "!/",
%o A244656                  d[st] - 1, "!"));
%o A244656             ret = s; break(2),
%o A244656             if(pr > s, st++; pr = d[st]; i = 0)),
%o A244656          if(pr < s, st += i + 1, st++); pr = d[st]; i = 0)))));
%o A244656 return(ret)}
%Y A244656 Cf. A003418, A000142, A025527, A002378.
%K A244656 nonn
%O A244656 1,1
%A A244656 _Rick L. Shepherd_, Jul 03 2014, Sep 14 2014