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A330507 a(n) is the smallest number k having for every prime p <= prime(n) at least one prime partition with least part p, and no such partition having least part > prime(n). If no such k exists then a(n) = 0 (see comments).

%I A330507 #43 Mar 16 2020 09:16:49
%S A330507 2,6,10,18,24,30,51,57,69,60,99,111,123,143,147,159,177,189,201,213,
%T A330507 225,245,255,267,291,303,309,321,345,357,381,393,411,427,447,465,471,
%U A330507 493,507,519,537,553,573,583,623,621,633,669,681,695,707,729,749,753,783
%N A330507 a(n) is the smallest number k having for every prime p <= prime(n) at least one prime partition with least part p, and no such partition having least part > prime(n). If no such k exists then a(n) = 0 (see comments).
%C A330507 Alternatively, a(n) is the smallest number whose product of distinct least part primes from all partitions of n into prime parts, is equal to primorial(n).
%C A330507 2 is the only prime term.
%C A330507 a(n) = 0 for n = 90, 151, 349, 352, 444, ... . - _Alois P. Heinz_, Mar 12 2020
%H A330507 Alois P. Heinz, <a href="/A330507/b330507.txt">Table of n, a(n) for n = 1..500</a>
%H A330507 Michael De Vlieger, <a href="/A333238/a333238.png">Labeled plot of p at (pi(p), n)</a>, for 2 <= n <= 240, with k in a(n) shown in red.
%e A330507 a(1) = 2 because [2] is the only prime partition of prime(1) = 2.
%e A330507 a(2) = 6 because [2,2,2] and [3,3] are the only possible prime partitions of 6, namely with prime(1) and prime(2) the only least parts.
%p A330507 b:= proc(n, p, t) option remember; `if`(n=0, 1, `if`(p>n, 0, (q->
%p A330507       add(b(n-p*j, q, 1), j=1..n/p)*t^p+b(n, q, t))(nextprime(p))))
%p A330507     end:
%p A330507 a:= proc(n) option remember; local f, k, p; p:= ithprime(n);
%p A330507       for k to 4*p do f:= b(k, 2, x); if degree(f)<= p and andmap(
%p A330507         h->0<coeff(f, x, h), [ithprime(i)$i=1..n]) then return k fi
%p A330507       od; 0
%p A330507     end:
%p A330507 seq(a(n), n=1..55);  # _Alois P. Heinz_, Mar 12 2020
%t A330507 With[{s = Array[Union@ Select[IntegerPartitions[#], AllTrue[#, PrimeQ] &][[All, -1]] &, 70]}, TakeWhile[Map[FirstPosition[s, #][[1]] &, Rest@ NestList[Append[#, Prime[Length@ # + 1]] &, {}, 12]], IntegerQ]] (* _Michael De Vlieger_, Mar 06 2020 *)
%t A330507 (* Second program: *)
%t A330507 Block[{a, m = 125, s}, a = ConstantArray[{}, m]; s = {Prime@ PrimePi@ m}; Do[If[# <= m, If[FreeQ[a[[#]], Last@ s], a = ReplacePart[a, # -> Union@ Append[a[[#]], Last@ s]], Nothing]; AppendTo[s, Last@ s], If[Last@ s == 2, s = DeleteCases[s, 2]; If[Length@ s == 0, Break[], s = MapAt[Prime[PrimePi[#] - 1] &, s, -1]], s = MapAt[Prime[PrimePi[#] - 1] &, s, -1]]] &@ Total[s], {i, Infinity}]; TakeWhile[Map[FirstPosition[a, #][[1]] &, Rest@ NestList[Append[#, Prime[Length@ # + 1]] &, {}, Max[Length /@ a]]], IntegerQ]] (* _Michael De Vlieger_, Mar 11 2020 *)
%Y A330507 Cf. A000040, A051034, A331634, A332861, A002110, A333238.
%K A330507 nonn
%O A330507 1,1
%A A330507 _David James Sycamore_, Mar 01 2020
%E A330507 a(19)-a(21) from _Michael De Vlieger_, Mar 12 2020
%E A330507 a(22)-a(55) from _Alois P. Heinz_, Mar 12 2020