A356603 Position in A356226 of first appearance of the n-th composition in standard order (row n of A066099).

1, 2, 4, 10, 8, 20, 50, 110, 16, 40, 100, 220, 250, 550, 1210, 1870, 32, 80, 200, 440, 500, 1100, 2420, 3740, 1250, 2750, 6050, 9350, 13310, 20570, 31790, 43010, 64, 160, 400, 880, 1000, 2200, 4840, 7480, 2500, 5500, 12100, 18700, 26620, 41140, 63580, 86020

Offset: 0

Author

Gus Wiseman, Aug 30 2022

Keywords

• nonn

Comments

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.

The image consists of all numbers whose prime indices are odd and cover an initial interval of odd positive integers.

Examples

			The terms together with their prime indices begin:
      1: {}
      2: {1}
      4: {1,1}
     10: {1,3}
      8: {1,1,1}
     20: {1,1,3}
     50: {1,3,3}
    110: {1,3,5}
     16: {1,1,1,1}
     40: {1,1,1,3}
    100: {1,1,3,3}
    220: {1,1,3,5}
    250: {1,3,3,3}
    550: {1,3,3,5}
   1210: {1,3,5,5}
   1870: {1,3,5,7}
		

Links

• Gus Wiseman, Statistics, classes, and transformations of standard compositions

Crossrefs

See link for sequences related to standard compositions.

The partitions with these Heinz numbers are counted by A053251.

A subset of A066208 (numbers with all odd prime indices).

Up to permutation, these are the positions of first appearances of rows in A356226. Other statistics are:

- length: A287170, firsts A066205

- minimum: A356227

- maximum: A356228

- bisected length: A356229

- standard composition: A356230

- Heinz number: A356231

The sorted version is A356232.

An ordered version is counted by A356604.

A001221 counts distinct prime factors, sum A001414.

A073491 lists numbers with gapless prime indices, complement A073492.

Cf. A000005, A001222, A055932, A061395, A073493, A132747, A137921, A193829, A286470, A356224, A356237.

Programs

Mathematica

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
stcinv[q_]:=1/2 Total[2^Accumulate[Reverse[q]]];
mnrm[s_]:=If[Min@@s==1,mnrm[DeleteCases[s-1,0]]+1,0];
sq=stcinv/@Table[Length/@Split[primeMS[n],#1>=#2-1&],{n,1000}];
Table[Position[sq,k][[1,1]],{k,0,mnrm[Rest[sq]]}]