A357861 - OEIS

%I #9 Oct 20 2022 12:44:21

%S 1,2,3,4,5,7,8,9,11,12,13,16,17,19,23,24,25,27,29,31,32,37,40,41,43,

%T 45,47,48,49,53,59,61,63,64,67,71,73,79,80,81,83,89,96,97,101,103,107,

%U 109,112,113,121,125,127,128,131,135,137,139,144,149,151,157

%N Numbers whose prime indices have weakly decreasing run-sums. Heinz numbers of the partitions counted by A304406.

%C 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.

%C 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.

%C 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).

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

%e The terms together with their prime indices begin:

%e     1: {}

%e     2: {1}

%e     3: {2}

%e     4: {1,1}

%e     5: {3}

%e     7: {4}

%e     8: {1,1,1}

%e     9: {2,2}

%e    11: {5}

%e    12: {1,1,2}

%e    13: {6}

%e    16: {1,1,1,1}

%e    17: {7}

%e    19: {8}

%e    23: {9}

%e    24: {1,1,1,2}

%e    25: {3,3}

%e    27: {2,2,2}

%e For example, the prime indices of 24 are {1,1,1,2}, with run-sums (3,2), which are weakly decreasing, so 24 is in the sequence.

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

%t Select[Range[100],GreaterEqual@@Total/@Split[primeMS[#]]&]

%Y These partitions are counted by A304406.

%Y These are the indices  of rows in A354584 that are weakly decreasing.

%Y The complement is A357850, counted by A357865, opposite A357876.

%Y The strictly decreasing version is A357864, counted by A304430.

%Y The opposite (weakly increasing) version is A357875, counted by A304405.

%Y A001222 counts prime factors, distinct A001221.

%Y A056239 adds up prime indices, row sums of A112798.

%Y Cf. A047966, A118914, A181819, A239312, A300273, A304442.

%K nonn

%O 1,2

%A _Gus Wiseman_, Oct 19 2022