login
Numbers whose prime indices do not have weakly decreasing run-sums. Heinz numbers of the partitions counted by A357865.
4

%I #5 Oct 20 2022 12:44:11

%S 6,10,14,15,18,20,21,22,26,28,30,33,34,35,36,38,39,42,44,46,50,51,52,

%T 54,55,56,57,58,60,62,65,66,68,69,70,72,74,75,76,77,78,82,84,85,86,87,

%U 88,90,91,92,93,94,95,98,99,100,102,104,105,106,108,110,111

%N Numbers whose prime indices do not have weakly decreasing run-sums. Heinz numbers of the partitions counted by A357865.

%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 6: {1,2}

%e 10: {1,3}

%e 14: {1,4}

%e 15: {2,3}

%e 18: {1,2,2}

%e 20: {1,1,3}

%e 21: {2,4}

%e 22: {1,5}

%e 26: {1,6}

%e 28: {1,1,4}

%e 30: {1,2,3}

%e 33: {2,5}

%e 34: {1,7}

%e 35: {3,4}

%e 36: {1,1,2,2}

%e 38: {1,8}

%e 39: {2,6}

%e 42: {1,2,4}

%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 are the indices of rows in A354584 that are not weakly decreasing.

%Y The complement is A357861, counted by A304406.

%Y These partitions are counted by A357865.

%Y The opposite (not weakly increasing) version is A357876, counted by A357878.

%Y A001222 counts prime factors, distinct A001221.

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

%Y Cf. A118914, A181819, A300273, A304405, A304428, A304430, A304442, A353832, A353864, A353932, A357864, A357875.

%K nonn

%O 1,1

%A _Gus Wiseman_, Oct 19 2022