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A265112 a(n) = A023360(A000040(n)): number of compositions of prime(n) into prime parts.

%I A265112 #14 Dec 16 2015 14:00:31
%S A265112 1,1,3,6,20,46,232,501,2352,24442,53243,550863,2616338,5701553,
%T A265112 27077005,280237217,2900328380,6320545915,65414893802,310664269401,
%U A265112 677015556295,7006815193063,33276323565116,344395408399372,7767597342090622,36889382062795742
%N A265112 a(n) = A023360(A000040(n)): number of compositions of prime(n) into prime parts.
%H A265112 Robert Israel, <a href="/A265112/b265112.txt">Table of n, a(n) for n = 1..768</a>
%e A265112 prime(4) = 7; a(4) = A023360(7) = 6 because there are 6 compositions of 7 into prime parts {2,3,5,7}: {7}, {5+2}, {3+2+2}, {2+5}, {2+3+2} and {2+2+3}.
%p A265112 N:= 1000: # to get a(1) to a(A000720(N))
%p A265112 Primes:= select(isprime, [2,seq(i,i=3..N,2)]):
%p A265112 M:= nops(Primes);
%p A265112 F:= proc(x)
%p A265112 option remember;
%p A265112 local k;
%p A265112 add(procname(x-Primes[k]),k=1..numtheory:-pi(x));
%p A265112 end proc:
%p A265112 F(0):= 1:
%p A265112 seq(F(Primes[n]),n=1..M); # _Robert Israel_, Dec 02 2015
%t A265112 Needs["Combinatorica`"]; Table[Length@ Flatten[Permutations[#, {Length@ #}] & /@ Select[Combinatorica`Partitions@ Prime@ n, AllTrue[#, PrimeQ] &], 1], {n, 14}] (* Version 10, slow, or *)
%t A265112 lim = 101; t = Rest@ CoefficientList[Series[1/(1 - Sum[x^Prime[i], {i, 1, PrimePi@ lim}]), {x, 0, lim}], x]; t[[#]] &@ Prime@ Range@ PrimePi@ lim
%t A265112 (* _Michael De Vlieger_, Dec 01 2015, after _David W. Wilson_ at A023360 *)
%Y A265112 Cf. A000040 (prime numbers), A023360.
%K A265112 nonn,easy
%O A265112 1,3
%A A265112 _Bob Selcoe_, Dec 01 2015