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

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, 31, 13, 11, 17, 11, 17, 37, 19, 13, 17, 41, 19, 43, 13, 13, 23, 47, 19, 13, 19, 17, 23, 53, 23, 17, 19, 19, 29, 59, 29, 61, 31, 17, 23, 19, 29, 67, 31, 23, 29, 71

Offset: 2

Author

David James Sycamore, Jan 23 2020

Keywords

• nonn

Examples

			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].
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 2..12500

Crossrefs

Cf. A000040, A000607, A001414, A100484, A001748.

Programs

Maple

b:= proc(n, p, t) option remember; `if`(n=0, 1, `if`(p>n, 0, (q->
      add(b(n-p*j, q, 1), j=1..n/p)*t^p+b(n, q, t))(nextprime(p))))
    end:
a:= n-> degree(b(n, 2, x)):
seq(a(n), n=2..100);  # Alois P. Heinz, Mar 13 2020

Mathematica

Array[If[PrimeQ@ #, #, Max@ IntegerPartitions[#, #/FactorInteger[#][[1, 1]], Prime@ Range@ PrimePi[# - 2]][[All, -1]] ] &, 60, 2] (* Michael De Vlieger, Jan 26 2020 *)
(* Second program: *)
b[n_, p_, t_] := b[n, p, t] = If[n == 0, 1, If[p > n, 0, Function[q, Sum[
     b[n - p*j, q, 1], {j, 1, n/p}]*t^p + b[n, q, t]][NextPrime[p]]]];
a[n_] := Exponent[b[n, 2, x], x];
a /@ Range[2, 100] (* Jean-François Alcover, Jun 04 2021, after Alois P. Heinz *)

Formula

For prime p>2, a(p) = a(2*p) = a(3*p) = p.