A284463 - OEIS

%I #15 Apr 22 2021 04:45:12

%S 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 15778,1,1,2,2,2,10252,1,2,2,1601,1,302265,1,80,750,2,1,299426,1,

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

%H Alois P. Heinz, <a href="/A284463/b284463.txt">Table of n, a(n) for n = 0..2000</a>

%H <a href="/index/Com#comp">Index entries for sequences related to compositions</a>

%F a(n) = [x^n] 1/(1 - Sum_{p|n, p prime} x^p).

%F a(n) = 1 if n is a prime power > 1.

%F a(n) = 2 if n is a squarefree semiprime.

%e 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 a:= proc(n) option remember; local b, l;

%p       l, b:= numtheory[factorset](n),

%p       proc(m) option remember; `if`(m=0, 1,

%p          add(`if`(j>m, 0, b(m-j)), j=l))

%p       end; b(n)

%p     end:

%p seq(a(n), n=0..100);  # _Alois P. Heinz_, Mar 28 2017

%t 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 (Python)

%o from sympy import divisors, isprime

%o from sympy.core.cache import cacheit

%o @cacheit

%o def a(n):

%o     l=[x for x in divisors(n) if isprime(x)]

%o     @cacheit

%o     def b(m): return 1 if m==0 else sum(b(m - j) for j in l if j <= m)

%o     return b(n)

%o print([a(n) for n in range(101)]) # _Indranil Ghosh_, Aug 01 2017, after Maple code

%Y Cf. A000040, A014652, A023360, A066882, A100346.

%K nonn

%O 0,7

%A _Ilya Gutkovskiy_, Mar 27 2017