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A331634 a(n) is the greatest possible least part of any prime partition of n. 9
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 (list; graph; refs; listen; history; text; internal format)
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-Fran├žois Alcover, Jun 04 2021, after Alois P. Heinz *)

CROSSREFS

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

Sequence in context: A338668 A284695 A081812 * A341676 A139421 A219964

Adjacent sequences:  A331631 A331632 A331633 * A331635 A331636 A331637

KEYWORD

nonn

AUTHOR

David James Sycamore, Jan 23 2020

STATUS

approved

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Last modified May 17 23:07 EDT 2022. Contains 353779 sequences. (Running on oeis4.)