A333365 -id:A333365 - cp's OEIS frontend

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

Michael De Vlieger, David James Sycamore, Mar 12 2020

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)

			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
...

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.

Cf. A330507, A333129, A333259, A333365.

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

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]

A335106 Irregular triangle T(n,k) is the number of times that prime(k) is the greatest part in a partition of n into prime parts; Triangle T(n,k), n>=0, 1 <= k <= max(1,A000720(A335285(n))), read by rows.

Original entry on oeis.org

0, 0, 1, 0, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 0, 2, 1, 1, 1, 1, 2, 1, 0, 2, 2, 1, 1, 1, 2, 2, 2, 0, 2, 3, 2, 1, 1, 1, 2, 3, 3, 1, 0, 3, 4, 3, 1, 1, 1, 2, 4, 4, 2, 1, 0, 3, 5, 5, 2, 1, 1, 1, 3, 5, 5, 3, 2, 0, 3, 6, 7, 3, 2, 1, 1, 1, 3, 7, 7, 4, 3, 1, 0, 4

Offset: 0

David James Sycamore, Michael De Vlieger, May 23 2020

Let L(n) be the set of least part primes of all prime partitions of n, and let G(n) be corresponding set of greatest prime parts. All prime partitions, commencing with p in L(n) and terminating with q in G(n) can be shown as unique pathways on a partition tree of n; see link for details. |G(n)| = A000607(n).

			A000607(10) = 5 and the prime partitions of 10 are: (2,2,2,2,2), (2,2,3,3), (2,3,5), (5,5) and (3,7). Thus G(10) = {2,3,5,5,7}, and consequently row 10 is [1,1,2,1]. In the table below, for n >= 2,  0 is used to indicate when prime(k) is not in G(n) and is less than the greatest member of G(n), otherwise the entry for prime(k) not in G(n) is left empty. For n >= 2 the sum of entries in the n-th row is |G(n)| = A000607(n). Triangle T(n,k) begins:
0;
0;
1;
0, 1;
1;
0, 1, 1;
1, 1;
0, 1, 1, 1;
1, 1, 1;
0, 2, 1, 1;
1, 1, 2, 1;
0, 2, 2, 1, 1;
1, 2, 2, 2;
0, 2, 3, 2, 1, 1;
1, 2, 3, 3, 1;
0, 3, 4, 3, 1, 1;
1, 2, 4, 4, 2, 1;
0, 3, 5, 5, 2, 1, 1;
...

David James Sycamore, Prime Partition Trees

Cf. A000040, A000607, A000720, A333365, A331634, A335285.
Row sums gives A000607 for n > 1. Length of n-th row is A000720(A335285(n)) for n >1.
Number of partition of n in the first k primes: A059841 (k = 1), A103221 (k = 2), A025795 (k = 3), A029144 (k = 4), A140952 (k = 5), A140953 (k = 6).

Flatten@ Block[{nn = 22, t}, t = Block[{s = {Prime@ PrimePi@ nn}}, KeySort@ Merge[#, Identity] &@ Join[{0 -> {}, 1 -> {}}, Reap[Do[If[# <= nn, Sow[# -> s]; 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}]][[-1, -1]] ] ]; Array[Function[p, If[! IntegerQ@ First@ p, {0}, Array[Count[p, Prime@ #] &, PrimePi@ Max@ p]]]@ Map[Max, t[[#]]] &, Max@ Keys@ t]] (* Michael De Vlieger, May 23 2020 *)
row[0]={0}; row[k_] := Join[If[OddQ@k, {0}, {}], Last /@ Tally@ Sort[ First /@ IntegerPartitions[k, All, Prime@ Range@ PrimePi@ k]]]; Join @@ Array[row, 20, 0] (* Giovanni Resta, May 31 2020 *)

More terms from Giovanni Resta, May 31 2020

A330433 Numbers k such that if there is a prime partition of k with least part p, then there exists at least one other prime partition of k with least part p.

Original entry on oeis.org

63, 161, 195, 235, 253, 425, 513, 581, 611, 615, 635, 667, 767, 779, 791, 803, 959, 1001, 1015, 1079, 1095, 1121, 1127, 1251, 1253, 1265, 1267, 1547, 1557, 1595, 1617, 1625, 1647, 1649, 1681, 1683, 1687, 1771, 1817, 1829, 1915, 1921, 2071, 2125, 2159, 2185

Offset: 1

David James Sycamore, Mar 01 2020

nonn

If k is prime then [k] is the only prime partition of k with least part k, and therefore k cannot be in this sequence. If k > 2 is even, then (assuming the validity of Goldbach's conjecture) there is a prime partition [p,q] of k (p <= q) in which p is the greatest possible least part and therefore no other partition of k is possible with least part p, so k is not a term. Therefore all terms of this sequence are odd composites.

			9 is not a term because [3,3,3] is the only prime partition of 9 having 3 as least part.
63 is a term because every possible prime partition is accounted for as follows, where (m,p) means m partitions of 63 with least part p: (2198,2), (323,3), (60,5), (15,7), (5,11), (2,13), (2,17), (sum of m values = 2605 = A000607(63)). 63 must be in the sequence because (1,p) does not appear in this list, and is the smallest such number because every odd composite < 63 has at least one prime partition with unique least part (as for 9 above).

Cf. A000040, A000607, A051034, A331634, A332861, A333365.

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:= proc(n) option remember; local k; for k from a(n-1)+1
      while 1 in {coeffs(b(k, 2, x))} do od; k
    end: a(0):=1:
seq(a(n), n=1..40);  # Alois P. Heinz, Mar 21 2020

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[0] = 1;
a[n_] := a[n] = Module[{k}, For[k = a[n-1]+1, MemberQ[CoefficientList[b[k, 2, x], x], 1], k++]; k];
Table[Print[n, " ", a[n]]; a[n], {n, 1, 40}] (* Jean-François Alcover, Nov 26 2020, after Alois P. Heinz *)

A333417 a(n) is the greatest number k having for every prime <= prime(n) at least one prime partition with least part p, and no such partition having least part > prime(n). If no such k exists then a(n) = 0.

Original entry on oeis.org

4, 9, 16, 27, 35, 49, 63, 65, 85, 95, 105, 121, 135, 145, 169, 175, 187, 203, 207, 221, 253, 265, 273, 289, 301, 305, 319, 351, 369, 387, 403, 407, 425, 445, 473, 485, 495, 517, 529, 545, 551, 567, 611, 615, 629, 637, 671, 679, 693, 697, 725, 747, 781, 793, 799

Offset: 1

David James Sycamore, Mar 20 2020

nonn

Alternatively a(n) is the greatest number whose product of distinct least part primes from all prime partitions of n, is equal to primorial(n). Companion sequence to A330507.
From Michael De Vlieger, Mar 20 2020: (Start)
a(n) = 0 for n = {90, 151, 349, 352, 444, ...}, cf. the comment from Alois P. Heinz at A330507.
Index m of last instance of A002110(n) in A333129 as m increases.
Last row n in A333238 that contains the consecutive primes (1...n).
Last index of the occurrence of 2^n - 1 in A333259, which is the decimal value of the characteristic function of primes in A333238 interpreted as a binary number. (End)

			a(1) = 4 because [2,2] is the only prime partition of 4, and no greater number n has only 2 as least part in any partition of n into primes.
From _Michael De Vlieger_, Mar 20 2020: (Start)
Looking at this sequence as the first position of 2^n - 1 in A333259, which in binary is a k-bit repunit, we look for the last occasion of such in A333259, indicated by the arrows. a(k) = n for rows n that have an arrow. In the chart, we reverse the portrayal of the binary rendition of A333259(n), replacing zeros with "." for clarity:
   n   A333259(n)            k
------------------------------
   2   1                     1
   3   . 1
   4   1                  -> 1
   5   1 . 1
   6   1 1                   2
   7   1 . . 1
   8   1 1                   2
   9   1 1                -> 2
  10   1 1 1                 3
  11   1 1 . . 1
  12   1 1 1                 3
  13   1 1 . . . 1
  14   1 1 . 1
  15   1 1 1                 3
  16   1 1 1              -> 3
  17   1 1 1 . . . 1
  18   1 1 1 1               4
  19   1 1 1 . . . . 1
  20   1 1 1 1               4
  ... (End)

Cf. A000040, A002110, A051034, A331634, A332861, A330507, A333129, A333238, A333259, A333365.

With[{s = TakeWhile[Import["https://oeis.org/A333259/b333259.txt", "Data"], Length@ # > 0 &][[All, -1]]}, Array[If[Length[#] == 0, 0, #[[-1, 1]] - 1] &@ Position[s, 2^# - 1] &, 55]] (* Michael De Vlieger, Mar 20 2020, using the b-file at A333259 *)

More terms from Michael De Vlieger, Mar 20 2020

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.

Original entry on oeis.org

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

David James Sycamore, Mar 30 2020

nonn

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.

			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)

Cf. A001043, A331634, A333238, A333365.

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 *)

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

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A335106 Irregular triangle T(n,k) is the number of times that prime(k) is the greatest part in a partition of n into prime parts; Triangle T(n,k), n>=0, 1 <= k <= max(1,A000720(A335285(n))), read by rows.

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A330433 Numbers k such that if there is a prime partition of k with least part p, then there exists at least one other prime partition of k with least part p.

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A333417 a(n) is the greatest number k having for every prime <= prime(n) at least one prime partition with least part p, and no such partition having least part > prime(n). If no such k exists then a(n) = 0.

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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.

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