%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