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A284463 Number of compositions (ordered partitions) of n into prime divisors of n.

%I A284463 #15 Apr 22 2021 04:45:12
%S A284463 1,0,1,1,1,1,2,1,1,1,2,1,12,1,2,2,1,1,65,1,23,2,2,1,351,1,2,1,38,1,
%T A284463 15778,1,1,2,2,2,10252,1,2,2,1601,1,302265,1,80,750,2,1,299426,1,
%U A284463 13404,2,107,1,1618192,2,5031,2,2,1,707445067,1,2,2398,1,2,119762253,1,173,2,39614048,1,255418101,1,2,154603
%N A284463 Number of compositions (ordered partitions) of n into prime divisors of n.
%H A284463 Alois P. Heinz, <a href="/A284463/b284463.txt">Table of n, a(n) for n = 0..2000</a>
%H A284463 <a href="/index/Com#comp">Index entries for sequences related to compositions</a>
%F A284463 a(n) = [x^n] 1/(1 - Sum_{p|n, p prime} x^p).
%F A284463 a(n) = 1 if n is a prime power > 1.
%F A284463 a(n) = 2 if n is a squarefree semiprime.
%e A284463 a(6) = 2 because 6 has 4 divisors {1, 2, 3, 6} among which 2 are primes {2, 3} therefore we have [3, 3] and [2, 2, 2].
%p A284463 a:= proc(n) option remember; local b, l;
%p A284463       l, b:= numtheory[factorset](n),
%p A284463       proc(m) option remember; `if`(m=0, 1,
%p A284463          add(`if`(j>m, 0, b(m-j)), j=l))
%p A284463       end; b(n)
%p A284463     end:
%p A284463 seq(a(n), n=0..100);  # _Alois P. Heinz_, Mar 28 2017
%t A284463 Table[d = Divisors[n]; Coefficient[Series[1/(1 - Sum[Boole[PrimeQ[d[[k]]]] x^d[[k]], {k, Length[d]}]), {x, 0, n}], x, n], {n, 0, 75}]
%o A284463 (Python)
%o A284463 from sympy import divisors, isprime
%o A284463 from sympy.core.cache import cacheit
%o A284463 @cacheit
%o A284463 def a(n):
%o A284463     l=[x for x in divisors(n) if isprime(x)]
%o A284463     @cacheit
%o A284463     def b(m): return 1 if m==0 else sum(b(m - j) for j in l if j <= m)
%o A284463     return b(n)
%o A284463 print([a(n) for n in range(101)]) # _Indranil Ghosh_, Aug 01 2017, after Maple code
%Y A284463 Cf. A000040, A014652, A023360, A066882, A100346.
%K A284463 nonn
%O A284463 0,7
%A A284463 _Ilya Gutkovskiy_, Mar 27 2017