A252665 Number of ways to write n as n = a*b*c*d*e with 1 <= a <= b <= c <= d <= e <= 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, 7, 2, 2, 2, 9, 1, 2, 2, 7, 1, 5, 1, 4, 4, 2, 1, 12, 2, 4, 2, 4, 1, 7, 2, 7, 2, 2, 1, 11, 1, 2, 4, 10, 2, 5, 1, 4, 2, 5, 1, 16, 1, 2, 4, 4, 2, 5, 1, 12, 5, 2, 1, 11, 2

Offset: 1

Author

Michel Lagneau, Dec 20 2014

Keywords

• nonn

Comments

Starts the same as, but is different from A218320 where a(n) = A218320(n) for n = 1..31. First values of n such that a(n) differs from A218320(n) are 32, 48, 64, 72, 80, ... .

Also starts the same as A001055, but differs from it for n = 64, ...

Examples

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

Links

• Michael De Vlieger, Table of n, a(n) for n = 1..10000
• Michael De Vlieger, Records and first positions of records

Crossrefs

Cf. A001055, A034836, A218320.

Programs

Maple

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, 5)
    end:
seq(a(n), n=1..100);  # Alois P. Heinz, Aug 31 2017

Mathematica

Table[c=0; Do[If[i<=j<=k<=l<=m && i*j*k*l*m==n, c++], {i, t=Divisors[n]}, {j, t}, {k, t}, {l, t}, {m, t}]; c, {n, 90}]
(* Second program: *)
b[n_, i_, t_] := b[n, i, t] = If[n == 1, 1, If[t == 1, Boole[n <= i], Sum[b[n/d, d, t - 1], {d, Select[Divisors@ n, # <= i &]}]]]; Parallelize@ Array[b[#, #, 5] &@ Apply[Times, Power @@@ Sort[FactorInteger[#], #1[[2]] > #2[[2]] &]] &, 120] (* Michael De Vlieger, Aug 31 2017, after Jean-François Alcover at A218320 *)