cp's OEIS Frontend

A218320 Number of ways to write n as n = a*b*c*d with 1 <= a <= b <= c <= d <= n.

%I A218320 #15 Mar 22 2017 11:17:40
%S A218320 1,1,1,2,1,2,1,3,2,2,1,4,1,2,2,5,1,4,1,4,2,2,1,7,2,2,3,4,1,5,1,6,2,2,
%T A218320 2,9,1,2,2,7,1,5,1,4,4,2,1,11,2,4,2,4,1,7,2,7,2,2,1,11,1,2,4,9,2,5,1,
%U A218320 4,2,5,1,15,1,2,4,4,2,5,1,11,5,2,1,11,2
%N A218320 Number of ways to write n as n = a*b*c*d with 1 <= a <= b <= c <= d <= n.
%C A218320 Starts the same as, but is different from A001055. First values of n such that a(n) differs from A001055(n) are 32, 48, 64, 72, 80, ... .
%C A218320 The value of a is the same for all numbers n with the same prime signature.  For prime p we have a(p^n) = A001400(n), the number of partitions of n into at most 4 parts. - _Alois P. Heinz_, Nov 03 2012
%H A218320 Alois P. Heinz, <a href="/A218320/b218320.txt">Table of n, a(n) for n = 1..10000</a>
%e A218320 a(12) = 4 because we can write 12 = 1*1*1*12 = 1*1*2*6 = 1*1*3*4 = 1*2*2*3.
%p A218320 for n from 1 to 90 do:t1:=0: for a from 1 to n do: for b from a to n do :for c from b to n do : for d from c to n do :if a*b*c*d = n then t1:=t1+1: else fi: od: od: od: od:printf(`%d, `,t1):od:
%p A218320 # second Maple program
%p A218320 with(numtheory):
%p A218320 b:= proc(n, i, t) option remember;
%p A218320       `if`(n=1, 1, `if`(t=1, `if`(n<=i, 1, 0),
%p A218320        add(b(n/d, d, t-1), d=select(x->x<=i, divisors(n)))))
%p A218320     end:
%p A218320 a:= proc(n) local l, m;
%p A218320       l:= sort(ifactors(n)[2], (x, y)-> x[2]>y[2]);
%p A218320       m:= mul(ithprime(i)^l[i][2], i=1..nops(l));
%p A218320       b(m, m, 4)
%p A218320     end:
%p A218320 seq(a(n), n=1..100);  # _Alois P. Heinz_, Nov 03 2012
%t A218320 b[n_, i_, t_] := b[n, i, t] = If[n==1, 1, If[t==1, If[n <= i, 1, 0], Sum[b[n/d, d, t-1], {d, Select[Divisors[n], # <= i&]}]]];
%t A218320 a[n_] := (l = Sort[FactorInteger[n], #1[[2]] > #2[[2]]&]; m = Times @@ Power @@@ l; b[m, m, 4]);
%t A218320 Array[a, 100] (* _Jean-François Alcover_, Mar 22 2017, after _Alois P. Heinz_ *)
%Y A218320 Cf. A001055, A034836.
%K A218320 nonn
%O A218320 1,4
%A A218320 _Michel Lagneau_, Oct 25 2012