A333636 a(n) is the greatest least part of a partition of n into prime parts which does not divide n, or 0 if no such prime exists.

0, 0, 0, 2, 0, 2, 3, 2, 3, 3, 5, 3, 3, 2, 5, 5, 7, 5, 7, 5, 5, 5, 11, 7, 7, 7, 11, 7, 13, 7, 13, 7, 11, 11, 17, 11, 7, 11, 17, 11, 19, 13, 13, 13, 17, 13, 19, 13, 19, 13, 23, 17, 23, 17, 19, 17, 17, 17, 29, 19, 19, 17, 23, 19, 29, 19, 31, 19, 29, 19, 31, 19, 31, 23, 29, 23, 37, 19, 37, 23, 29, 23, 41

Offset

2,4

Comments

For n = 2,3,4,6 a(n) = 0. For n > 6 there are no terms a(n) = 0, and therefore n has at least one prime partition whose least part prime does not divide n. This sequence lists the greatest such prime for each n. The indices of the records of this sequence are in A001043.

Links

Table of n, a(n) for n=2..84.

Example

The only prime partition of 2 is [2], but 2|2, so a(2) = 0. Also, since [2,2,2] and [3,3] are the prime partitions of 6, with 2|6 and 3|6, a(6) = 0. The prime partitions of 5 are [2,3] and [5], but 2 does not divide 5 so a(5) = 2.
From Michael De Vlieger, Apr 01 2020: (Start)
Chart showing terms k in rows 5 <= n <= 24 of A333238, plotted at pi(k), with "." replacing terms k | n. In the table, we place a(n) in parenthesis:
   n   k
  -------------------
   5  (2)      .
   6   .   .
   7  (2)          .
   8   .  (3)
   9  (2)  .
  10   .  (3)  .
  11   2  (3)          .
  12   .   .  (5)
  13   2  (3)              .
  14   .  (3)      .
  15  (2)  .   .
  16   .   3  (5)
  17   2   3  (5)              .
  18   .   .   5  (7)
  19   2   3  (5)                  .
  20   .   3   .  (7)
  21   2   .  (5)  .
  22   .   3  (5)      .
  23   2   3  (5)                      .
  24   .   .   5   7  (11)
  ... (End)

Mathematica

Block[{m = 84, s, a}, a = ConstantArray[{}, m]; s = {Prime@ PrimePi@ m}; Do[If[# <= m, If[And[FreeQ[a[[#]], Last[s]], Mod[#, Last[s]] != 0], 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}]; Map[If[Length[#] == 0, 0, Last@ #] &, Rest@ a]] (* Michael De Vlieger, Apr 01 2020 *)

Crossrefs

Cf. A001043, A331634, A333238, A333365.

Sequence in context: A212184 A371571 A033769 * A074660 A002125 A171731

Adjacent sequences: A333633 A333634 A333635 * A333637 A333638 A333639

Keyword

nonn

Author

David James Sycamore, Mar 30 2020

Status

approved