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Number of ways of writing n = m + f! with m a semiprime and f > 0.
2

%I #9 Sep 20 2024 23:55:00

%S 0,0,0,0,1,1,1,1,0,2,2,2,0,0,2,3,1,0,0,1,1,1,2,1,0,1,3,3,0,1,1,1,1,2,

%T 2,2,1,1,3,3,2,0,0,1,2,1,1,1,1,2,1,2,1,0,1,1,3,2,3,1,1,1,3,2,0,1,1,1,

%U 0,2,2,0,1,0,3,1,0,1,2,1,1,1,2,1,0,2,2,3,2,0,1,2,3,1,2,2,2,1,1,1,2,0,0,0,0

%N Number of ways of writing n = m + f! with m a semiprime and f > 0.

%C Number of partitions of n into a semiprime and a factorial.

%H Robert Israel, <a href="/A181676/b181676.txt">Table of n, a(n) for n = 1..10000</a>

%e a(16)=3 because 16 = 2*5 + 3! = 2*7 + 2! = 3*5 + 1!.

%p N:= 200: # for a(1) .. a(N)

%p R:= Vector(N):

%p S:= select(t -> numtheory:-bigomega(t)=2, [$1..N]):

%p F:= 1: v:= 1:

%p for i from 2 do v:= v*i; if v > N then break fi; F:= F,v od:

%p F:= [F]:

%p for s in S do

%p for f in F do

%p v:= s+f;

%p if v > N then break fi;

%p R[v]:= R[v]+1

%p od od:

%p convert(R,list); # _Robert Israel_, Sep 20 2024

%Y Cf. A000142, A001358, A175940.

%K nonn

%O 1,10

%A _Juri-Stepan Gerasimov_, Nov 04 2010