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Number of compositions (ordered partitions) of the n-th prime into distinct prime parts.
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%I #10 Nov 26 2020 11:35:37

%S 1,1,3,3,1,3,25,9,61,91,99,151,901,303,1759,3379,5239,4713,8227,12901,

%T 12537,23059,65239,159421,232369,489817,351237,726295,564363,1101883,

%U 2517865,6916027,11825821,4942227,27166753,21280053,39547957,52630273,113638975

%N Number of compositions (ordered partitions) of the n-th prime into distinct prime parts.

%H Alois P. Heinz, <a href="/A331901/b331901.txt">Table of n, a(n) for n = 1..1000</a>

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

%F a(n) = A219107(A000040(n)).

%e a(4) = 3 because we have [7], [5, 2] and [2, 5].

%p s:= proc(n) option remember; `if`(n<1, 0, ithprime(n)+s(n-1)) end:

%p b:= proc(n, i, t) option remember; `if`(s(i)<n, 0, `if`(n=0, t!, (p

%p ->`if`(p>n, 0, b(n-p, i-1, t+1)))(ithprime(i))+b(n, i-1, t)))

%p end:

%p a:= n-> b(ithprime(n), n, 0):

%p seq(a(n), n=1..42); # _Alois P. Heinz_, Jan 31 2020

%t s[n_] := s[n] = If[n < 1, 0, Prime[n] + s[n - 1]];

%t b[n_, i_, t_] := b[n, i, t] = If[s[i] < n, 0, If[n == 0, t!, Function[p, If[p > n, 0, b[n - p, i - 1, t + 1]]][Prime[i]] + b[n, i - 1, t]]];

%t a[n_] := b[Prime[n], n, 0];

%t Array[a, 42] (* _Jean-François Alcover_, Nov 26 2020, after _Alois P. Heinz_ *)

%Y Cf. A000040, A023360, A056768, A070215, A219107, A265112, A299168.

%K nonn

%O 1,3

%A _Ilya Gutkovskiy_, Jan 31 2020