A357862 Numbers whose prime indices have strictly increasing run-sums. Heinz numbers of the partitions counted by A304428.

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 41, 42, 43, 44, 46, 47, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 61, 62, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74

Offset

1,2

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 Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.

The sequence of runs of a sequence consists of its maximal consecutive constant subsequences when read left-to-right. For example, the runs of (2,2,1,1,1,3,2,2) are (2,2), (1,1,1), (3), (2,2), with sums (4,3,3,4).

Links

Table of n, a(n) for n=1..67.

Mathematics Stack Exchange, What is a sequence run? (answered 2011-12-01)

Example

The terms together with their prime indices begin:
    1: {}
    2: {1}
    3: {2}
    4: {1,1}
    5: {3}
    6: {1,2}
    7: {4}
    8: {1,1,1}
    9: {2,2}
   10: {1,3}
   11: {5}
   13: {6}
   14: {1,4}
   15: {2,3}
   16: {1,1,1,1}
   17: {7}
   18: {1,2,2}
   19: {8}
For example, the prime indices of 24 are {1,1,1,2}, with run-sums (3,2), which are not strictly increasing, so 24 is not in the sequence.

Mathematica

primeMS[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
Select[Range[100], Less@@Total/@Split[primeMS[#]]&]

Crossrefs

These partitions are counted by A304428.

The complement is A357863.

These are the indices of rows in A354584 that are strictly increasing.

The opposite (strictly decreasing) version is A357864, counted by A304430.

The weakly increasing version is A357875, counted by A304405.

A001222 counts prime factors, distinct A001221.

A056239 adds up prime indices, row sums of A112798.

Cf. A118914, A181819, A275870, A300273, A304442.

Sequence in context: A130572 A353838 A183224 * A098240 A023805 A168186

Adjacent sequences: A357859 A357860 A357861 * A357863 A357864 A357865

Keyword

nonn

Author

Gus Wiseman, Oct 19 2022

Status

approved