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A344789 Number of partitions of the n-th nonprime number into a nonprime number of nonprime parts.

%I A344789 #16 Sep 08 2022 07:50:46
%S A344789 1,2,2,2,4,3,6,8,11,11,19,27,32,37,55,63,78,88,108,149,204,232,274,
%T A344789 313,371,497,556,654,864,1135,1267,1476,1915,2142,2474,2754,3182,4070,
%U A344789 4528,5190,5769,6594,8347,10530,11666,13240,14657,16597,20747,22924,25854
%N A344789 Number of partitions of the n-th nonprime number into a nonprime number of nonprime parts.
%H A344789 Alois P. Heinz, <a href="/A344789/b344789.txt">Table of n, a(n) for n = 1..5000</a>
%e A344789 a(5) = 4: [9], [6,1,1,1], [4,1,1,1,1,1], [1,1,1,1,1,1,1,1,1].
%e A344789 a(6) = 3: [10], [4,4,1,1], [1,1,1,1,1,1,1,1,1,1].
%e A344789 a(7) = 6: [12], [9,1,1,1], [6,4,1,1], [4,4,1,1,1,1], [4,1,1,1,1,1,1,1,1], [1,1,1,1,1,1,1,1,1,1,1,1].
%p A344789 c:= proc(n) option remember; local k; if n=1 then 1 else
%p A344789       for k from 1+c(n-1) while isprime(k) do od; k fi
%p A344789     end:
%p A344789 h:= proc(n) option remember; `if`(isprime(n), h(n-1), n) end:
%p A344789 b:= proc(n, i, c) option remember; `if`(n=0 or i=1, `if`(isprime(
%p A344789       c+n), 0, 1), b(n-i, h(min(n-i, i)), c+1)+b(n, h(i-1), c))
%p A344789     end:
%p A344789 a:= n-> b(c(n)$2, 0):
%p A344789 seq(a(n), n=1..55);
%t A344789 c[n_] := c[n] = Module[{k}, If[n == 1, 1,
%t A344789    For[k = 1+c[n-1], PrimeQ[k], k++]; k]];
%t A344789 h[n_] := h[n] = If[PrimeQ[n], h[n-1], n];
%t A344789 b[n_, i_, c_] := b[n, i, c] = If[n == 0 || i == 1, If[PrimeQ[
%t A344789    c+n], 0, 1], b[n-i, h[Min[n-i, i]], c+1] + b[n, h[i-1], c]];
%t A344789 a[n_] := b[c[n], c[n], 0];
%t A344789 Table[a[n], {n, 1, 55}] (* _Jean-François Alcover_, Sep 08 2022, after _Alois P. Heinz_ *)
%Y A344789 Cf. A018252, A316154.
%K A344789 nonn
%O A344789 1,2
%A A344789 _Alois P. Heinz_, May 28 2021