A252665 - OEIS

%I #25 Dec 26 2018 03:41:43

%S 1,1,1,2,1,2,1,3,2,2,1,4,1,2,2,5,1,4,1,4,2,2,1,7,2,2,3,4,1,5,1,7,2,2,

%T 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,

%U 4,2,5,1,16,1,2,4,4,2,5,1,12,5,2,1,11,2

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

%C 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, ... .

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

%H Michael De Vlieger, <a href="/A252665/b252665.txt">Table of n, a(n) for n = 1..10000</a>

%H Michael De Vlieger, <a href="/A252665/a252665.txt">Records and first positions of records</a>

%e 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.

%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, 5)

%p     end:

%p seq(a(n), n=1..100);  # _Alois P. Heinz_, Aug 31 2017

%t 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}]

%t (* Second program: *)

%t 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 *)

%Y Cf. A001055, A034836, A218320.

%K nonn

%O 1,4

%A _Michel Lagneau_, Dec 20 2014