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%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
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