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

2,1

Links

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

Formula

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

Example

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].

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 *)

Crossrefs

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

Sequence in context: A284695 A081812 A356427 * A370833 A341676 A139421

Adjacent sequences: A331631 A331632 A331633 * A331635 A331636 A331637

Keyword

nonn

Author

David James Sycamore, Jan 23 2020

Status

approved