A333238 Irregular table where row n lists the distinct smallest primes p of prime partitions of n.

2, 3, 2, 2, 5, 2, 3, 2, 7, 2, 3, 2, 3, 2, 3, 5, 2, 3, 11, 2, 3, 5, 2, 3, 13, 2, 3, 7, 2, 3, 5, 2, 3, 5, 2, 3, 5, 17, 2, 3, 5, 7, 2, 3, 5, 19, 2, 3, 5, 7, 2, 3, 5, 7, 2, 3, 5, 11, 2, 3, 5, 23, 2, 3, 5, 7, 11, 2, 3, 5, 7, 2, 3, 5, 7, 13, 2, 3, 5, 7, 2, 3, 5, 7, 11

Offset: 2

Author

Michael De Vlieger, David James Sycamore, Mar 12 2020

Keywords

• nonn
• tabf

Comments

A prime partition of n is an integer partition wherein all parts are prime. For instance, (3 + 2) is a prime partition of the sum 5; for n = 5, (5) is also a prime partition. For 6, we have two prime partitions (3 + 3) and (2 + 2 + 2).

We note that there are no prime partitions for n = 1, therefore the offset of this sequence is 2.

The number of prime partitions of n is shown by A000607(n).

For prime p, row p includes p itself as the largest term, since p is the sum of (p).

The product of all terms in row n gives A333129(n). - Alois P. Heinz, Mar 16 2020

From David James Sycamore, Mar 28 2020: (Start)

In the irregular table below, T(n,k) is either prime(k) or is empty. The former means there is at least one prime partition of n with least part prime(k), the latter means that no such partition exists. T(n,k) empty is not recorded in the data.

Recursion for n >= 4: T(n,k) = prime(k) iff T(n-prime(k), k) = prime(k), or there is a q > k such that T(n-prime(k), q) = prime(q); else T(n,k) is empty. Example: T(17,3) = 5 because T(12,3) = prime(3) = 5. T(10,2) = 3 since although T(7,2) is empty, T(7,4) = prime(4) = 7. (End)

Examples

			The least primes among the prime partitions of 5 are 2 and 5, cf. the 2 prime partitions of 5: (5) and (3, 2), thus row 5 lists {2, 5}.
The least primes among the prime partitions of 6 are 2 and 3, cf. the two prime partitions of 6, (3, 3), and (2, 2, 2), thus row 6 lists {2, 3}.
Row 7 contains {2, 7} because there are 3 prime partitions of 7: (7), (5, 2), (3, 2, 2). Note that 2 is the smallest part of the latter two partitions, thus only 2 and 7 are distinct.
Table plotting prime p in row n at pi(p) place, intervening primes missing from row n are shown by "." as a place holder:
n      Primes in row n
----------------------
2:     2
3:     .   3
4:     2
5:     2   .   5
6:     2   3
7:     2   .   .   7
8:     2   3
9:     2   3
10:    2   3   5
11:    2   3   .   .  11
12:    2   3   5
13:    2   3   .   .   .  13
14:    2   3   .   7
15:    2   3   5
16:    2   3   5
17:    2   3   5   .   .   .  17
...
		

Links

• Alois P. Heinz, Rows n = 2..1000, flattened (rows n=2..240 from Michael De Vlieger)
• Michael De Vlieger, Labeled plot of p at (pi(p), n), for 2 <= n <= 240, with terms n in A330507 shown in red.

Crossrefs

Cf. A330507, A333129, A333259, A333365.

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:
T:= proc(n) option remember; (p-> seq(`if`(isprime(i) and
      coeff(p, x, i)>0, i, [][]), i=2..degree(p)))(b(n, 2, x))
    end:
seq(T(n), n=2..40);  # Alois P. Heinz, Mar 16 2020

Mathematica

Block[{a, m = 20, s}, a = ConstantArray[{}, m]; s = {Prime@ PrimePi@ m}; Do[If[# <= m, If[FreeQ[a[[#]], First@ s], a = ReplacePart[a, # -> 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}]; Union /@ a // Flatten]