A357877 The a(n)-th composition in standard order is the sequence of run-sums of the prime indices of n.

0, 1, 2, 2, 4, 6, 8, 4, 8, 12, 16, 10, 32, 24, 20, 8, 64, 24, 128, 20, 40, 48, 256, 18, 32, 96, 32, 40, 512, 52, 1024, 16, 80, 192, 72, 40, 2048, 384, 160, 36, 4096, 104, 8192, 80, 68, 768, 16384, 34, 128, 96, 320, 160, 32768, 96, 144, 72, 640, 1536, 65536, 84

Offset: 1

Author

Gus Wiseman, Oct 17 2022

Keywords

• nonn

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 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).

The k-th composition in standard order (graded reverse-lexicographic, A066099) is obtained by taking the set of positions of 1's in the reversed binary expansion of k, prepending 0, taking first differences, and reversing again. This gives a bijective correspondence between nonnegative integers and integer compositions.

Examples

			The prime indices of 24 are (1,1,1,2), with run-sums (3,2), and this is the 18th composition in standard order, so a(24) = 18.
		

Links

• Mathematics Stack Exchange, What is a sequence run? (answered 2011-12-01)

Crossrefs

The version for prime indices instead of standard compositions is A353832.

The version for standard compositions instead of prime indices is A353847.

A ranking of the rows of A354584.

A001222 counts prime factors, distinct A001221.

A011782 counts compositions.

A047966 counts uniform partitions, compositions A329738.

A056239 adds up prime indices, row sums of A112798.

A066099 lists standard compositions.

A351014 counts distinct runs in standard compositions.

Cf. A118914, A181819, A238279, A239312, A275870, A300273, A304405, A304442, A304660, A333755, A353743-A354912, A357875.

Programs

Mathematica

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
stcinv[q_]:=Total[2^(Accumulate[Reverse[q]])]/2;
Table[stcinv[Total/@Split[primeMS[n]]],{n,100}]