A357864 Numbers whose prime indices have strictly decreasing run-sums. Heinz numbers of the partitions counted by A304430.

1, 2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17, 19, 23, 24, 25, 27, 29, 31, 32, 37, 41, 43, 45, 47, 48, 49, 53, 59, 61, 64, 67, 71, 73, 79, 80, 81, 83, 89, 96, 97, 101, 103, 107, 109, 113, 121, 125, 127, 128, 131, 135, 137, 139, 149, 151, 157, 160, 163, 167, 169, 173

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..62.

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}
    7: {4}
    8: {1,1,1}
    9: {2,2}
   11: {5}
   13: {6}
   16: {1,1,1,1}
   17: {7}
   19: {8}
   23: {9}
   24: {1,1,1,2}
   25: {3,3}
   27: {2,2,2}
   29: {10}
For example, the prime indices of 24 are {1,1,1,2}, with run-sums (3,2), which are strictly decreasing, so 24 is in the sequence.

Mathematica

primeMS[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
Select[Range[300], Greater@@Total/@Split[primeMS[#]]&]

Crossrefs

Subsequence of A304686.

These partitions are counted by A304430.

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

The weakly decreasing version is A357861, counted by A304406.

The opposite version is A357862, counted by A304428, complement A357863.

A001222 counts prime factors, distinct A001221.

A056239 adds up prime indices, row sums of A112798.

Cf. A118914, A181819, A300273, A304405, A304442, A357875.

Sequence in context: A322902 A302040 A302036 * A331912 A326848 A328957

Adjacent sequences: A357861 A357862 A357863 * A357865 A357866 A357867

Keyword

nonn

Author

Gus Wiseman, Oct 19 2022

Status

approved