A381436 Irregular triangle read by rows where row k is the section-sum partition of the prime indices of n.

1, 2, 1, 1, 3, 3, 4, 1, 1, 1, 2, 2, 4, 5, 3, 1, 6, 5, 5, 1, 1, 1, 1, 7, 3, 2, 8, 4, 1, 6, 6, 9, 3, 1, 1, 3, 3, 7, 2, 2, 2, 5, 1, 10, 6, 11, 1, 1, 1, 1, 1, 7, 8, 7, 3, 3, 12, 9, 8, 4, 1, 1, 13, 7, 14, 6, 1, 5, 2, 10, 15, 3, 1, 1, 1, 4, 4, 4, 3, 9, 7, 1, 16, 3, 2, 2

Offset: 1

Author

Gus Wiseman, Feb 28 2025

Keywords

• nonn
• tabf

Comments

Row-lengths are A051903.

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

The section-sum partition of a multiset or partition y is defined as follows: (1) determine and remember the sum of all distinct parts, (2) remove one instance of each distinct part, (3) repeat until no parts are left. The remembered values comprise the section-sum partition. For example, starting with (3,2,2,1,1) we get (6,3).

Equivalently, the k-th part of the section-sum partition is the sum of all (distinct) parts that appear at least k times. Compare to the definition of the conjugate of a partition, where we count parts >= k.

The conjugate of a section-sum partition is a Look-and-Say partition; see A048767, union A351294, count A239455.

Examples

			The prime indices of 24 are (2,1,1,1), with sections ((2,1),(1),(1)), so row 24 is (3,1,1).
Triangle begins:
   1: (empty)
   2: 1
   3: 2
   4: 1 1
   5: 3
   6: 3
   7: 4
   8: 1 1 1
   9: 2 2
  10: 4
  11: 5
  12: 3 1
  13: 6
  14: 5
  15: 5
  16: 1 1 1 1
		

Crossrefs

Row-lengths are A051903.

Row sums are A056239.

First part in each row is A066328.

Taking length instead of sum gives A238744, Heinz numbers A238745, conjugate A181819.

Partitions of this type are counted by A239455, complement A351293.

Heinz numbers are A381431 (union A381432, complement A381433, fixed A000961, A000005).

Rows appearing only once have Heinz numbers A381434, more than once A381435.

Last part in each row is A381437, counted by A381438.

The conjugate is A381440, Heinz numbers A048767 (union A351294, complement A351295).

A000040 lists the primes.

A003963 gives product of prime indices.

A055396 gives least prime index, greatest A061395.

A056239 adds up prime indices, row sums of A112798.

A122111 represents conjugation in terms of Heinz numbers.

Set multipartitions: A050320, A089259, A116540, A270995, A296119, A318360, A318361.

Partition ideals: A300383, A317141, A381078, A381441, A381452, A381454.

Cf. A000720, A001222, A003557, A005117, A047966, A116861, A130091, A380955.

Programs

Mathematica

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
egs[y_]:=If[y=={},{},Table[Total[Select[Union[y],Count[y,#]>=i&]],{i,Max@@Length/@Split[y]}]];
Table[egs[prix[n]],{n,100}]