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A165933
Least integer, k, whose value is n in A165413.
5
1, 4, 35, 536, 16775, 1060976, 135007759, 34460631520, 17617985239071, 18027600169142208, 36907002795598798911, 151143401509104346210176, 1238053384151947477501575295, 20283338091738780737237428502272, 664629209970464486086782992577855743
OFFSET
1,2
COMMENTS
An alternative name: The smallest number whose binary expansion has exactly n distinct run-lengths. - Gus Wiseman, Feb 21 2022
Term a(n) has one 1, followed by n 0's, then two 1's, (n-1) 0's, ..., up to n runs; see Python program. - Michael S. Branicky, Feb 22 2022
LINKS
Michael S. Branicky, Table of n, a(n) for n = 1..81
EXAMPLE
a(1) in binary is 1, a(2) in binary is 100, a(3) in binary is 100011, a(4) in binary is 1000011000, etc.
From Gus Wiseman, Feb 21 2022: (Start)
The terms and their binary expansions begin:
n a(n)
1: 1 = 1
2: 4 = 100
3: 35 = 100011
4: 536 = 1000011000
5: 16775 = 100000110000111
6: 1060976 = 100000011000001110000
7: 135007759 = 1000000011000000111000001111
8: 34460631520 = 100000000110000000111000000111100000
9: 17617985239071 = 100000000011000000001110000000111100000011111
(End)
MATHEMATICA
g[n_] := Table[ {Table[1, {i}], Table[0, {n - i + 1}]}, {i, Floor[(n + If[ OddQ@n, 1, 0])/2]}]; f[n_] := FromDigits[ If[ OddQ@n, Flatten@ Most@ Flatten[ g@n, 1], Flatten@ g@n], 2]; Array[f, 14]
s=Table[Length[Union[Length/@Split[IntegerDigits[n, 2]]]], {n, 0, 1000}]; Table[Position[s, k][[1, 1]]-1, {k, Union[s]}] (* Gus Wiseman, Feb 21 2022 *)
PROG
(Python)
def a(n): # returns term by construction
if n == 1: return 1
q, r = divmod(n+1, 2)
s = "".join("1"*i + "0"*(n+1-i) for i in range(1, q+1))
if r == 0: s = s.rstrip("0")
return int(s, 2)
print([a(n) for n in range(1, 16)]) # Michael S. Branicky, Feb 22 2022
CROSSREFS
A subset of A044813 (distinct run-lengths) and of A175413 (distinct runs).
These are the positions of first appearances in A165413.
The version for runs instead of run-lengths is A350952, firsts of A297770.
A000120 counts binary weight.
A005811 counts runs in binary expansion.
A242882 counts compositions with distinct multiplicities.
A318928 gives runs-resistance of binary expansion.
A351014 counts distinct runs in standard compositions.
Counting words with all distinct run-lengths:
- A032020 = binary expansions, for runs A351018.
- A329739 = compositions, for runs A351013.
- A351017 = binary words, for runs A351016.
- A351292 = patterns, for runs A351200.
Sequence in context: A171778 A119391 A177387 * A005973 A007134 A334412
KEYWORD
easy,nonn
AUTHOR
Robert G. Wilson v, Sep 30 2009
EXTENSIONS
a(15) and beyond from Michael S. Branicky, Feb 22 2022
STATUS
approved