OFFSET
0,2
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.
LINKS
EXAMPLE
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}
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]]}]
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:
- 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.
KEYWORD
nonn
AUTHOR
Gus Wiseman, Aug 30 2022
STATUS
approved