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A331901 Number of compositions (ordered partitions) of the n-th prime into distinct prime parts.

%I A331901 #10 Nov 26 2020 11:35:37
%S A331901 1,1,3,3,1,3,25,9,61,91,99,151,901,303,1759,3379,5239,4713,8227,12901,
%T A331901 12537,23059,65239,159421,232369,489817,351237,726295,564363,1101883,
%U A331901 2517865,6916027,11825821,4942227,27166753,21280053,39547957,52630273,113638975
%N A331901 Number of compositions (ordered partitions) of the n-th prime into distinct prime parts.
%H A331901 Alois P. Heinz, <a href="/A331901/b331901.txt">Table of n, a(n) for n = 1..1000</a>
%H A331901 <a href="/index/Com#comp">Index entries for sequences related to compositions</a>
%F A331901 a(n) = A219107(A000040(n)).
%e A331901 a(4) = 3 because we have [7], [5, 2] and [2, 5].
%p A331901 s:= proc(n) option remember; `if`(n<1, 0, ithprime(n)+s(n-1)) end:
%p A331901 b:= proc(n, i, t) option remember; `if`(s(i)<n, 0, `if`(n=0, t!, (p
%p A331901       ->`if`(p>n, 0, b(n-p, i-1, t+1)))(ithprime(i))+b(n, i-1, t)))
%p A331901     end:
%p A331901 a:= n-> b(ithprime(n), n, 0):
%p A331901 seq(a(n), n=1..42);  # _Alois P. Heinz_, Jan 31 2020
%t A331901 s[n_] := s[n] = If[n < 1, 0, Prime[n] + s[n - 1]];
%t A331901 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 A331901 a[n_] := b[Prime[n], n, 0];
%t A331901 Array[a, 42] (* _Jean-François Alcover_, Nov 26 2020, after _Alois P. Heinz_ *)
%Y A331901 Cf. A000040, A023360, A056768, A070215, A219107, A265112, A299168.
%K A331901 nonn
%O A331901 1,3
%A A331901 _Ilya Gutkovskiy_, Jan 31 2020