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Subsequence of A304686.

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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.

The complement is... ?

The opposite (strictly increasing) weakly decreasing version is A357862, A357861, counted by A304428, complement A357863A304406.

The weak version (weakly decreasing) opposite version is A357861, A357862, counted by A304406A304428, complement A357863.

Cf. A118914 pri_sig_sort, A181819 pri_shadow, A275870 collapsible, A300273 h_collapsible, A304405 runsums_nondec, A304442 ptns_w_eq_runsums, A353743-A354912, A357875 prix_runsums_nondec.

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

allocated for Gus WisemanNumbers 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

1,2

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).

Mathematics Stack Exchange, <a href="https://math.stackexchange.com/q/87559">What is a sequence run? (answered 2011-12-01)</a>

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}

primeMS[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];

Select[Range[300], Greater@@Total/@Split[primeMS[#]]&]

These partitions are counted by A304430.

The complement is... ?

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

The opposite (strictly increasing) version is A357862, counted by A304428, complement A357863.

The weak version (weakly decreasing) version is A357861, counted by A304406.

A001222 counts prime factors, distinct A001221.

A056239 adds up prime indices, row sums of A112798.

Cf. A118914 pri_sig_sort, A181819 pri_shadow, A275870 collapsible, A300273 h_collapsible, A304405 runsums_nondec, A304442 ptns_w_eq_runsums, A353743-A354912, A357875 prix_runsums_nondec.

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Gus Wiseman, Oct 19 2022

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