A357863 - OEIS

%I #5 Oct 20 2022 12:45:03

%S 12,24,40,45,48,60,63,80,84,90,96,112,120,126,132,135,144,156,160,168,

%T 175,180,189,192,204,224,228,240,252,264,270,275,276,280,288,297,300,

%U 312,315,320,325,336,348,350,351,352,360,372,378,384,405,408,420,440

%N Numbers whose prime indices do not have strictly increasing run-sums. Heinz numbers of the partitions not counted by A304428.

%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    12: {1,1,2}

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

%e    40: {1,1,1,3}

%e    45: {2,2,3}

%e    48: {1,1,1,1,2}

%e    60: {1,1,2,3}

%e    63: {2,2,4}

%e    80: {1,1,1,1,3}

%e    84: {1,1,2,4}

%e    90: {1,2,2,3}

%e    96: {1,1,1,1,1,2}

%e   112: {1,1,1,1,4}

%e   120: {1,1,1,2,3}

%e   126: {1,2,2,4}

%e   132: {1,1,2,5}

%e   135: {2,2,2,3}

%e   144: {1,1,1,1,2,2}

%e   156: {1,1,2,6}

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

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

%Y These are the indices  of rows in A354584 that are not strictly increasing.

%Y The complement (strictly increasing) is A357862, counted by A304428.

%Y The weak (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, A304430, A304442, A357864, A357875.

%K nonn

%O 1,1

%A _Gus Wiseman_, Oct 19 2022