This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A190940 #34 Nov 23 2024 09:32:01
%S A190940 1,1,2,2,6,4,12,8,16,18,48,32,96,72,64,48,240,128,480,288,320,384,960,
%T A190940 512,960,1152,960,1152,3840,3072,7680,3072,6912,7680,6144,6144,18432,
%U A190940 15360,13824,12288,36864,23040,73728,49152,49152,61440,147456,73728,147456,122880
%N A190940 Number of divisors of LCM(1,2,...,n)/n.
%C A190940 Also, number of sequences of d1 = 1 < d2 < ... < dk = n for some k >= 1 that are the first k divisors of some integer (cf. A378314). - _Max Alekseyev_, Nov 22 2024
%C A190940 Also, the number of distinct values taken by lcm(a,a+b,a+b+c,...,n), where positive integers a,b,c,... run over the compositions a+b+c+...=n. - Conjectured by _Ridouane Oudra_, Aug 24 2019; proved by _Max Alekseyev_, Nov 22 2024
%C A190940 Proof. It is clear that n | lcm(a,a+b,...,n) | lcm(1,2,...,n). Hence, lcm(a,a+b,...,n) = d*n for some d | lcm(1,2,...,n)/n. We'll show that each such d is achievable. Suppose d*n has prime factorization p1^e1 * ... * pk^ek with p1^e1 < ... < pk^ek. It is clear that pk^ek <= n, and we can take a composition (a,b,c,...) = (p1^e1, p2^e2 - p1^e1, p3^e3 - p2^e2, ..., pk^ek - p(k-1)^e(k-1), n - pk^ek), which delivers lcm(a,a+b,a+b+c,...,n) = p1^e1 * ... * pk^ek = d*n. QED - _Max Alekseyev_, Nov 22 2024
%F A190940 a(n) = A000005(A002944(n)).
%e A190940 Examples: for n=3 the a(3) = 2 distinct values are 3, 6. The compositions are 3, 1+2, 2+1, and 1+1+1. The values of the lcm are lcm(3)=3, lcm(1,1+2)=3, lcm(2,2+1)=6, and lcm(1,1+1,1+1+1)=6.
%p A190940 Lpsum := proc(L) local ps,k ; ps := [op(1,L)] ; for i from 2 to nops(L) do ps := [op(ps), op(-1,ps)+op(i,L)] ; end do: ps ; end proc:
%p A190940 A190940 := proc(n) local lc,k,c ; lc := {} ; for k from 1 to n do for c in combinat[composition](n,k) do lc := lc union { ilcm( op(Lpsum(c))) }; end do: end do: nops(lc) ; end proc: # _R. J. Mathar_, Jun 02 2011
%t A190940 a[n_] := LCM @@@ (Accumulate /@ (Permutations /@ Rest[IntegerPartitions[n]] // Flatten[#, 1]&)) // Union // Length; Table[Print[an = a[n]]; an, {n, 1, 24}] (* _Jean-François Alcover_, Feb 27 2014 *)
%Y A190940 First difference of A378314.
%Y A190940 Cf. A000005, A002944, A003418, A056793, A101207.
%K A190940 nonn
%O A190940 1,3
%A A190940 _Naohiro Nomoto_, May 24 2011
%E A190940 a(12)-a(20) from _R. J. Mathar_, Jun 02 2011
%E A190940 a(21)-a(24) from _Alois P. Heinz_, Nov 03 2011
%E A190940 Edited and terms a(25) onward added by _Max Alekseyev_, Nov 22 2024