A381437 Last part of the section-sum partition of the prime indices of n.

0, 1, 2, 1, 3, 3, 4, 1, 2, 4, 5, 1, 6, 5, 5, 1, 7, 2, 8, 1, 6, 6, 9, 1, 3, 7, 2, 1, 10, 6, 11, 1, 7, 8, 7, 3, 12, 9, 8, 1, 13, 7, 14, 1, 2, 10, 15, 1, 4, 3, 9, 1, 16, 2, 8, 1, 10, 11, 17, 1, 18, 12, 2, 1, 9, 8, 19, 1, 11, 8, 20, 1, 21, 13, 3, 1, 9, 9, 22, 1, 2

Offset: 1

Author

Gus Wiseman, Feb 28 2025

Keywords

• nonn

Comments

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 (A381436) 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 972 are {1,1,2,2,2,2,2}, with section-sum partition (3,3,2,2,2), so a(972) = 2.
		

Crossrefs

Positions of first appearances are A008578.

The length of this partition is A051903.

The conjugate version is A051904.

For first instead of last part we get A066328.

These partitions are counted by A239455, complement A351293.

Positions of 1 are A360013, complement A381439.

This is the least prime index of A381431 (see A381432, A381433, A381434, A381435).

This is the last part of row n of A381436 (see A381440, A048767, A351294, A351295).

Counting partitions by this statistic gives A381438.

A000040 lists the primes, differences A001223.

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.

Cf. A000720, A001221, A001222, A003557, A047966, A116861, A130091, A181819, 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[If[n==1,0,Last[egs[prix[n]]]],{n,100}]

Formula

a(n) = A055396(A381431(n)).