cp's OEIS Frontend

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

%I A356603 #7 Aug 30 2022 09:41:41
%S A356603 1,2,4,10,8,20,50,110,16,40,100,220,250,550,1210,1870,32,80,200,440,
%T A356603 500,1100,2420,3740,1250,2750,6050,9350,13310,20570,31790,43010,64,
%U A356603 160,400,880,1000,2200,4840,7480,2500,5500,12100,18700,26620,41140,63580,86020
%N A356603 Position in A356226 of first appearance of the n-th composition in standard order (row n of A066099).
%C A356603 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 A356603 The image consists of all numbers whose prime indices are odd and cover an initial interval of odd positive integers.
%H A356603 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 A356603 The terms together with their prime indices begin:
%e A356603       1: {}
%e A356603       2: {1}
%e A356603       4: {1,1}
%e A356603      10: {1,3}
%e A356603       8: {1,1,1}
%e A356603      20: {1,1,3}
%e A356603      50: {1,3,3}
%e A356603     110: {1,3,5}
%e A356603      16: {1,1,1,1}
%e A356603      40: {1,1,1,3}
%e A356603     100: {1,1,3,3}
%e A356603     220: {1,1,3,5}
%e A356603     250: {1,3,3,3}
%e A356603     550: {1,3,3,5}
%e A356603    1210: {1,3,5,5}
%e A356603    1870: {1,3,5,7}
%t A356603 primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A356603 stcinv[q_]:=1/2 Total[2^Accumulate[Reverse[q]]];
%t A356603 mnrm[s_]:=If[Min@@s==1,mnrm[DeleteCases[s-1,0]]+1,0];
%t A356603 sq=stcinv/@Table[Length/@Split[primeMS[n],#1>=#2-1&],{n,1000}];
%t A356603 Table[Position[sq,k][[1,1]],{k,0,mnrm[Rest[sq]]}]
%Y A356603 See link for sequences related to standard compositions.
%Y A356603 The partitions with these Heinz numbers are counted by A053251.
%Y A356603 A subset of A066208 (numbers with all odd prime indices).
%Y A356603 Up to permutation, these are the positions of first appearances of rows in A356226. Other statistics are:
%Y A356603 - length: A287170, firsts A066205
%Y A356603 - minimum: A356227
%Y A356603 - maximum: A356228
%Y A356603 - bisected length: A356229
%Y A356603 - standard composition: A356230
%Y A356603 - Heinz number: A356231
%Y A356603 The sorted version is A356232.
%Y A356603 An ordered version is counted by A356604.
%Y A356603 A001221 counts distinct prime factors, sum A001414.
%Y A356603 A073491 lists numbers with gapless prime indices, complement A073492.
%Y A356603 Cf. A000005, A001222, A055932, A061395, A073493, A132747, A137921, A193829, A286470, A356224, A356237.
%K A356603 nonn
%O A356603 0,2
%A A356603 _Gus Wiseman_, Aug 30 2022