A218320 - OEIS

%I #15 Mar 22 2017 11:17:40

%S 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 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 4,2,5,1,15,1,2,4,4,2,5,1,11,5,2,1,11,2

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

%C 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 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 Alois P. Heinz, <a href="/A218320/b218320.txt">Table of n, a(n) for n = 1..10000</a>

%e 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 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 # second Maple program

%p with(numtheory):

%p b:= proc(n, i, t) option remember;

%p       `if`(n=1, 1, `if`(t=1, `if`(n<=i, 1, 0),

%p        add(b(n/d, d, t-1), d=select(x->x<=i, divisors(n)))))

%p     end:

%p a:= proc(n) local l, m;

%p       l:= sort(ifactors(n)[2], (x, y)-> x[2]>y[2]);

%p       m:= mul(ithprime(i)^l[i][2], i=1..nops(l));

%p       b(m, m, 4)

%p     end:

%p seq(a(n), n=1..100);  # _Alois P. Heinz_, Nov 03 2012

%t 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 a[n_] := (l = Sort[FactorInteger[n], #1[[2]] > #2[[2]]&]; m = Times @@ Power @@@ l; b[m, m, 4]);

%t Array[a, 100] (* _Jean-François Alcover_, Mar 22 2017, after _Alois P. Heinz_ *)

%Y Cf. A001055, A034836.

%K nonn

%O 1,4

%A _Michel Lagneau_, Oct 25 2012