A284463 -id:A284463 - cp's OEIS frontend

A300703 Number of compositions (ordered partitions) of n into prime parts that do not divide n.

1, 0, 0, 0, 0, 2, 0, 5, 2, 5, 2, 19, 2, 45, 6, 10, 14, 231, 4, 500, 14, 48, 45, 2351, 12, 1520, 144, 637, 100, 24441, 6, 53242, 810, 2558, 1294, 15402, 94, 550862, 3707, 16658, 680, 2616337, 53, 5701552, 11639, 6606, 30749, 27077004, 800, 21212965, 5215, 611097, 109818, 280237216

Offset: 0

Ilya Gutkovskiy, Mar 11 2018

nonn

			a(9) = 5 because we have [7, 2], [5, 2, 2], [2, 7], [2, 5, 2] and [2, 2, 5].

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Index entries for sequences related to compositions

Cf. A000040, A023360, A128515, A209402, A284463, A300702, A300704, A300706.

a:= proc(m) option remember; local b; b:= proc(n) option
      remember; `if`(n=0, 1, add(`if`(not isprime(j) or
       irem(m, j)=0, 0, b(n-j)), j=2..n)) end; b(m)
    end:
seq(a(n), n=0..80);  # Alois P. Heinz, Mar 11 2018

Table[SeriesCoefficient[1/(1 - Sum[Boole[Mod[n, k] != 0 && PrimeQ[k]] x^k, {k, 1, n}]), {x, 0, n}], {n, 0, 53}]

A286851 Number of compositions (ordered partitions) of n into unitary divisors of n.

Original entry on oeis.org

1, 1, 2, 2, 2, 2, 25, 2, 2, 2, 129, 2, 170, 2, 742, 450, 2, 2, 4603, 2, 1503, 3321, 29967, 2, 9278, 2, 200390, 2, 13460, 2, 154004511, 2, 2, 226020, 9262157, 51886, 127654, 2, 63346598, 2044895, 170354, 2, 185493291001, 2, 1304512, 567124, 2972038875, 2, 59489916, 2, 20367343494, 184947044, 14324735, 2

Offset: 0

Ilya Gutkovskiy, Aug 01 2017

nonn

			a(8) = 2 because 8 has 4 divisors {1, 2, 4, 8} among which 2 are unitary divisors {1, 8} therefore we have [8] and [1, 1, 1, 1, 1, 1, 1, 1].

Alois P. Heinz, Table of n, a(n) for n = 0..2000
Eric Weisstein's World of Mathematics, Unitary Divisor
Index entries for sequences related to compositions

Cf. A066874, A100346, A246655, A284463, A284464, A284465, A284466.

a:= proc(n) option remember; local b, l; l, b:=
      select(x-> igcd(x, n/x)=1, numtheory[divisors](n)),
      proc(m) option remember; `if`(m=0, 1,
         add(`if`(j>m, 0, b(m-j)), j=l))
      end; b(n)
    end:
seq(a(n), n=0..60);  # Alois P. Heinz, Aug 01 2017

Join[{1}, Table[d = Divisors[n]; Coefficient[Series[1/(1 - Sum[Boole[GCD[n/d[[k]], d[[k]]] == 1] x^d[[k]], {k, Length[d]}]), {x, 0, n}], x, n], {n, 1, 53}]]

from sympy import divisors, gcd
from sympy.core.cache import cacheit
@cacheit
def a(n):
    l=[x for x in divisors(n) if gcd(x, n//x)==1]
    @cacheit
    def b(m): return 1 if m==0 else sum(b(m - j) for j in l if j <= m)
    return b(n)
print([a(n) for n in range(61)]) # Indranil Ghosh, Aug 01 2017, after Maple code

a(n) = [x^n] 1/(1 - Sum_{d|n, gcd(d, n/d) = 1} x^d).

a(n) = 2 if n is a prime power (A246655).

cp's OEIS Frontend

A300703 Number of compositions (ordered partitions) of n into prime parts that do not divide n.

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