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Position in A356226 of first appearance of the n-th composition in standard order (row n of A066099).
12

%I #7 Aug 30 2022 09:41:41

%S 1,2,4,10,8,20,50,110,16,40,100,220,250,550,1210,1870,32,80,200,440,

%T 500,1100,2420,3740,1250,2750,6050,9350,13310,20570,31790,43010,64,

%U 160,400,880,1000,2200,4840,7480,2500,5500,12100,18700,26620,41140,63580,86020

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

%C 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.

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

%H Gus Wiseman, <a href="https://docs.google.com/document/d/e/2PACX-1vTCPiJVFUXN8IqfLlCXkgP15yrGWeRhFS4ozST5oA4Bl2PYS-XTA3sGsAEXvwW-B0ealpD8qnoxFqN3/pub">Statistics, classes, and transformations of standard compositions</a>

%e The terms together with their prime indices begin:

%e 1: {}

%e 2: {1}

%e 4: {1,1}

%e 10: {1,3}

%e 8: {1,1,1}

%e 20: {1,1,3}

%e 50: {1,3,3}

%e 110: {1,3,5}

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

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

%e 100: {1,1,3,3}

%e 220: {1,1,3,5}

%e 250: {1,3,3,3}

%e 550: {1,3,3,5}

%e 1210: {1,3,5,5}

%e 1870: {1,3,5,7}

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

%t stcinv[q_]:=1/2 Total[2^Accumulate[Reverse[q]]];

%t mnrm[s_]:=If[Min@@s==1,mnrm[DeleteCases[s-1,0]]+1,0];

%t sq=stcinv/@Table[Length/@Split[primeMS[n],#1>=#2-1&],{n,1000}];

%t Table[Position[sq,k][[1,1]],{k,0,mnrm[Rest[sq]]}]

%Y See link for sequences related to standard compositions.

%Y The partitions with these Heinz numbers are counted by A053251.

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

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

%Y - length: A287170, firsts A066205

%Y - minimum: A356227

%Y - maximum: A356228

%Y - bisected length: A356229

%Y - standard composition: A356230

%Y - Heinz number: A356231

%Y The sorted version is A356232.

%Y An ordered version is counted by A356604.

%Y A001221 counts distinct prime factors, sum A001414.

%Y A073491 lists numbers with gapless prime indices, complement A073492.

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

%K nonn

%O 0,2

%A _Gus Wiseman_, Aug 30 2022