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

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, 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, 4, 2, 5, 1, 15, 1, 2, 4, 4, 2, 5, 1, 11, 5, 2, 1, 11, 2

Offset: 1

Author

Michel Lagneau, Oct 25 2012

Keywords

• nonn

Comments

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, ... .

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

Examples

			a(12) = 4 because we can write 12 = 1*1*1*12 = 1*1*2*6 = 1*1*3*4 = 1*2*2*3.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A001055, A034836.

Programs

Maple

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:
# second Maple program
with(numtheory):
b:= proc(n, i, t) option remember;
      `if`(n=1, 1, `if`(t=1, `if`(n<=i, 1, 0),
       add(b(n/d, d, t-1), d=select(x->x<=i, divisors(n)))))
    end:
a:= proc(n) local l, m;
      l:= sort(ifactors(n)[2], (x, y)-> x[2]>y[2]);
      m:= mul(ithprime(i)^l[i][2], i=1..nops(l));
      b(m, m, 4)
    end:
seq(a(n), n=1..100);  # Alois P. Heinz, Nov 03 2012

Mathematica

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&]}]]];
a[n_] := (l = Sort[FactorInteger[n], #1[[2]] > #2[[2]]&]; m = Times @@ Power @@@ l; b[m, m, 4]);
Array[a, 100] (* Jean-François Alcover, Mar 22 2017, after Alois P. Heinz *)