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A331634 a(n) is the greatest possible least part of any prime partition of n.

%I A331634 #43 Jun 04 2021 04:42:05
%S A331634 2,3,2,5,3,7,3,3,5,11,5,13,7,5,5,17,7,19,7,7,11,23,11,7,13,7,11,29,13,
%T A331634 31,13,11,17,11,17,37,19,13,17,41,19,43,13,13,23,47,19,13,19,17,23,53,
%U A331634 23,17,19,19,29,59,29,61,31,17,23,19,29,67,31,23,29,71
%N A331634 a(n) is the greatest possible least part of any prime partition of n.
%H A331634 Alois P. Heinz, <a href="/A331634/b331634.txt">Table of n, a(n) for n = 2..12500</a>
%F A331634 For prime p>2, a(p) = a(2*p) = a(3*p) = p.
%e A331634 a(12) = 5, because 5 is the largest of all minimal primes in partitions of 12 into prime parts: [2,2,2,2,2,2], [2,2,2,3,3], [3,3,3,3], [2,2,3,5], [2,5,5], [2,3,7], [5,7].
%p A331634 b:= proc(n, p, t) option remember; `if`(n=0, 1, `if`(p>n, 0, (q->
%p A331634       add(b(n-p*j, q, 1), j=1..n/p)*t^p+b(n, q, t))(nextprime(p))))
%p A331634     end:
%p A331634 a:= n-> degree(b(n, 2, x)):
%p A331634 seq(a(n), n=2..100);  # _Alois P. Heinz_, Mar 13 2020
%t A331634 Array[If[PrimeQ@ #, #, Max@ IntegerPartitions[#, #/FactorInteger[#][[1, 1]], Prime@ Range@ PrimePi[# - 2]][[All, -1]] ] &, 60, 2] (* _Michael De Vlieger_, Jan 26 2020 *)
%t A331634 (* Second program: *)
%t A331634 b[n_, p_, t_] := b[n, p, t] = If[n == 0, 1, If[p > n, 0, Function[q, Sum[
%t A331634      b[n - p*j, q, 1], {j, 1, n/p}]*t^p + b[n, q, t]][NextPrime[p]]]];
%t A331634 a[n_] := Exponent[b[n, 2, x], x];
%t A331634 a /@ Range[2, 100] (* _Jean-François Alcover_, Jun 04 2021, after _Alois P. Heinz_ *)
%Y A331634 Cf. A000040, A000607, A001414, A100484, A001748.
%K A331634 nonn
%O A331634 2,1
%A A331634 _David James Sycamore_, Jan 23 2020