A037016 -id:A037016 - cp's OEIS frontend

A167755 Match the multisets in A175020 with those in A037016 then merge the two sequences.

1, 1, 2, 2, 3, 3, 5, 5, 4, 6, 7, 7, 10, 10, 12, 12, 9, 13, 8, 14, 15, 15, 21, 21, 19, 25, 18, 26, 24, 28, 17, 29, 16, 30, 31, 31, 42, 42, 36, 50, 51, 51, 37, 53, 56, 56, 35, 57, 34, 58, 48, 60, 33, 61, 32, 62, 63, 63

Offset: 1

Alford Arnold, Nov 10 2009

A175020 records the multiset with minimum value; whereas A037016 is based on reading binary expansion from right to left, run lengths increase.

			After the initial zero, A037016 begins:
  1
  2 3
  5 6 7
  10 12 13 14 15
  21 25 26 28 29 30 31
  42 50 51 53 56 57 58 60 61 62 63
  85 ...
and after resorting, A175020 begins:
  1
  2 3
  5 4 7
  10 12 9 8 15
  21 ...
so the irregular table begins:
  1 1
  2 2 3 3
  5 5 4 6 7 7
  10 10 12 12 9 13 8 14 15 15
  etc.
In binary, the number 9 maps to multiset (1,2,1) and the number 13 maps to (2,1,1), so 9 and 13 appear together in the sequence.

Cf. A000041 (1/2 row length of the irregular Table). A000975 (first & second column).

A125106 Enumeration of partitions by binary representation: each 1 is a part; the part size is 1 more than the number of 0's in the rest of the number.

Original entry on oeis.org

1, 2, 1, 1, 3, 2, 1, 2, 2, 1, 1, 1, 4, 3, 1, 3, 2, 2, 1, 1, 3, 3, 2, 2, 1, 2, 2, 2, 1, 1, 1, 1, 5, 4, 1, 4, 2, 3, 1, 1, 4, 3, 3, 2, 1, 3, 2, 2, 2, 1, 1, 1, 4, 4, 3, 3, 1, 3, 3, 2, 2, 2, 1, 1, 3, 3, 3, 2, 2, 2, 1, 2, 2, 2, 2, 1, 1, 1, 1, 1

Offset: 1

Alford Arnold, Dec 10 2006

Another way to describe this: starting with the binary representation and a counter set at one, count the 0's from right to left. Write a term equal to the counter for each "1" encountered.
A101211 is a similar sequence, with A005811 elements per row which maps natural numbers to compositions (ordered partitions).
There are two ways to consider this as a table: taking each partition as a row, or taking the partitions generated by 2^(n-1) through 2^n-1 as a row.
Taking the n-th row as multiple partitions, it consists of those partitions with the first hook size (largest part plus number of parts minus 1) equal to n. The number of integers in this n-th row is A001792(n-1), and the row sum is A049611.
Taking each partition as a separate row, the row lengths are A000120, and the row sums are A161511.
Heinz numbers of the rows are A005940. - Gus Wiseman, Jan 17 2023

			Row 4:
1000 [4]
1001 [3,1]
1010 [3,2]
1011 [2,1,1]
1100 [3,3]
1101 [2,2,1]
1110 [2,2,2]
1111 [1,1,1,1]

Alois P. Heinz, Rows n = 1..12, flattened

Cf. A000041, A005811, A037016, A101211, A001792, A049611, A126411.
Each partition as row: A000120 (row widths), A161511 (row sums), A243499 (row products).
Cf. A005940. - Franklin T. Adams-Watters, Mar 06 2010
Lasts are A001511.
Firsts are A008687.
Cf. A019565, A029837, A029931, A048793, A059893, A066099, A070939, A242628.

b:= proc(n) local c, l, m; l:=[][]; m:= n; c:=1;
      while m>0 do if irem(m, 2, 'm')=0 then c:= c+1
         else l:= c, l fi
      od; l
    end:
T:= n-> seq(b(i), i=2^(n-1)..2^n-1):
seq(T(n), n=1..7);  # Alois P. Heinz, Sep 25 2015

f[k_] := (bits = IntegerDigits[k, 2]; zerosCount = Reverse[ Accumulate[ 1-Reverse[bits] ] ] + 1; Select[ Transpose[ {bits, zerosCount} ], First[#] == 1 & ][[All, 2]]); row[n_] := Table[ f[k], {k, 2^(n-1), 2^n-1}]; Flatten[ Table[ row[n], {n, 1, 5}]] (* Jean-François Alcover, Jan 24 2012 *)
scc[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
Table[Reverse[scc[n]-Range[Length[scc[n]]]+1],{n,0,20}] (* Gus Wiseman, Jan 17 2023 *)

Partition 2n is partition n with every part size increased by 1; partition 2n+1 is partition n with an additional part of size 1.

T(n,k) = A272020(n,k) - A000120(n) + k. - Gus Wiseman, Jan 17 2023

Edited by Franklin T. Adams-Watters, Jun 11 2009

A037013 Reading binary expansion from right to left, run lengths strictly decrease.

Original entry on oeis.org

0, 1, 3, 4, 7, 8, 15, 16, 24, 31, 32, 39, 48, 63, 64, 79, 96, 112, 127, 128, 143, 159, 192, 224, 255, 256, 287, 319, 384, 399, 448, 480, 511, 512, 543, 575, 624, 639, 768, 799, 896, 960, 1023, 1024, 1087, 1151, 1248, 1279, 1536, 1567, 1599, 1792, 1920, 1984, 2047, 2048, 2111

Offset: 1

N. J. A. Sloane

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000
Index entries for sequences related to binary expansion of n

Subsequence of A037014, cf. A037015, A037016.

import Data.List (unfoldr, group)
a037013 n = a037013_list !! (n-1)
a037013_list = 0 : filter
   (all (< 0) . (\x -> zipWith (-) (tail $ rls x) $ rls x)) [1..] where
       rls = map length . group . unfoldr
             (\x -> if x == 0 then Nothing else Just $ swap $ divMod x 2)
-- Reinhard Zumkeller, Mar 10 2012

Select[Range[0,2200],Min[Differences[Length/@Split[ IntegerDigits[ #,2]]]]>0&] (* Harvey P. Dale, Dec 17 2012 *)

More terms from Patrick De Geest, Feb 15 1999

Offset fixed and missing 1024 and 2047 inserted by Reinhard Zumkeller, Mar 10 2012

A037015 Numbers n with property that, reading binary expansion of n from right to left, run lengths strictly increase.

Original entry on oeis.org

0, 1, 3, 6, 7, 14, 15, 28, 30, 31, 57, 60, 62, 63, 120, 121, 124, 126, 127, 241, 248, 249, 252, 254, 255, 483, 496, 497, 504, 505, 508, 510, 511, 966, 993, 995, 1008, 1009, 1016, 1017, 1020, 1022, 1023, 1987, 1990, 2016, 2017, 2019, 2032, 2033, 2040, 2041, 2044

Offset: 1

N. J. A. Sloane

There are A000009(k) elements of this list consisting of k bits. - Jason Kimberley, Jan 22 2013

			From _Jason Kimberley_, Jan 30 2013: (Start)
Interleaved lines:
binary expansions,
corresponding run lengths (distinct partitions);
1,
1;
11,
2;
110, 111,
2,1; 3;
1110, 1111,
3,1; 4;
11100, 11110, 11111,
3,2; 4,1; 5;
111001, 111100, 111110, 111111,
3,2,1; 4,2; 5,1; 6;
1111000, 1111001, 1111100, 1111110, 1111111,
4,3; 4,2,1; 5,2; 6,1; 7;
11110001, 11111000, 11111001, 11111100, 11111110, 11111111
4,3,1; 5,3; 5,2,1; 6,2; 7,1; 8;
111100011, 111110000, 111110001, 111111000, 111111001, 111111100, 111111110, 111111111,
4,3,2; 5,4; 5,3,1; 6,3; 6,2,1; 7,2; 8,1; 9;
Notice the reversed sorting when a part corresponds to a run of 0's.
(End)

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000
Index entries for sequences related to binary expansion of n

Subsequence of A037016, cf. A037013, A037014.

Cf. A030308.

import Data.List (group)
a037015 n = a037015_list !! (n-1)
a037015_list = filter (all (> 0) . ds) [0..] where
   ds x = zipWith (-) (tail gs) gs where
      gs = map length $ group $ a030308_row x
-- Reinhard Zumkeller, Jul 31 2013, Mar 10 2012

Select[Range[0,2500],Min[Differences[Length/@Split[ Reverse[ IntegerDigits[ #,2]]]]]>0&] (* Harvey P. Dale, Nov 18 2014 *)
Select[Range[0,2100],Max[Differences[Length/@Split[IntegerDigits[#,2]]]]<0&] (* Harvey P. Dale, Jun 28 2020 *)

from itertools import groupby
A037015_list = []
for n in range(10**5):
    c = None
    for x, y in groupby(bin(n)[2:]):
        z = len(list(y))
        if c != None and z >= c:
            break
        c = z
    else:
        A037015_list.append(n) # Chai Wah Wu, Sep 14 2021

More terms from Patrick De Geest, Feb 15 1999

Offset fixed and missing 1023 inserted by Reinhard Zumkeller, Mar 10 2012

A037014 Numbers n with property that reading binary expansion from right to left (from least significant to most significant), run lengths do not increase.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 7, 8, 10, 11, 12, 15, 16, 19, 20, 21, 23, 24, 31, 32, 39, 40, 42, 43, 44, 47, 48, 51, 56, 63, 64, 71, 76, 79, 80, 83, 84, 85, 87, 88, 95, 96, 103, 112, 127, 128, 143, 152, 159, 160, 167, 168, 170, 171, 172, 175, 176, 179, 184, 191, 192, 199, 204, 207

Offset: 1

N. J. A. Sloane

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Index entries for sequences related to binary expansion of n

Cf. A037013 (subsequence), A037016, A037015.

import Data.List (unfoldr, group)
a037014 n = a037014_list !! (n-1)
a037014_list = 0 : filter
   (all (<= 0) . (\x -> zipWith (-) (tail $ rls x) $ rls x)) [1..] where
       rls = map length . group . unfoldr
             (\x -> if x == 0 then Nothing else Just $ swap $ divMod x 2)
-- Reinhard Zumkeller, Mar 10 2012

Select[Range[0,250],Min[Differences[Length/@Split[IntegerDigits[ #,2]]]]>= 0&] (* Harvey P. Dale, Jan 30 2013 *)

More terms from Patrick De Geest, Feb 15 1999

Offset fixed by Reinhard Zumkeller, Mar 10 2012

A335835 Sort the run lengths in binary expansion of n in descending order (with multiplicities).

Original entry on oeis.org

0, 1, 2, 3, 6, 5, 6, 7, 14, 13, 10, 13, 12, 13, 14, 15, 30, 29, 26, 25, 26, 21, 26, 29, 28, 25, 26, 25, 28, 29, 30, 31, 62, 61, 58, 57, 50, 53, 50, 57, 58, 53, 42, 53, 50, 53, 58, 61, 60, 57, 50, 51, 50, 53, 50, 57, 56, 57, 58, 57, 60, 61, 62, 63, 126, 125

Offset: 0

Rémy Sigrist, Jun 26 2020

This sequence preserves the number of runs (A005811) and the length (A070939) of the binary representation of a number.

			For n = 72:
- the binary representation of 72 is "1001000",
- the corresponding run lengths are: 1, 2, 1, 3,
- in descending order: 3, 2, 1, 1,
- so the binary representation of a(72) is "1110010",
- and a(72) = 114.

Rémy Sigrist, Table of n, a(n) for n = 0..8191
Index entries for sequences related to binary expansion of n

Cf. A005811, A037016 (fixed points), A070939, A101211, A335834.

torl(n) = { my (rr=[]); while (n, my (r=valuation(n+(n%2), 2)); rr = concat(r, rr); n\=2^r); rr }
fromrl(rr) = { my (v=0); for (k=1, #rr, v = (v+(k%2))*2^rr[k]-(k%2)); v }
a(n) = { fromrl(vecsort(torl(n),,4)) }

a(a(n)) = a(n).

A331625 Numbers k such that both k and k+1 are exceptional (A072066).

Original entry on oeis.org

1215, 98415, 273375, 413343, 846368, 1987983, 2302911, 6082047, 6200144, 8089712, 9034496, 9861183, 11868848, 13010463, 13325391, 13955247, 16159743, 16592768, 17537552, 18482336, 20686832, 20883663, 21198591, 22143375, 22891328, 23206256, 24347871, 25607583

Offset: 1

Jianing Song, Jan 22 2020

nonn

Conjecture: for every p > 0, there exist infinitely many k such that k, k+1, ..., k+p-1 are all exceptional numbers. In specific, there exist infinitely many k such that both k and k+1 are exceptional.

			1215 = 3^5 * 5, 1216 = 2^6 * 19;
thus A037019(1215) = 2^4 * 3^2 * 5^2 * 7^2 * 11^2 * 13^2 = 3607203600, A037016(1216) = 2^18 * 3 * 5 * 7 * 11 * 13 * 17 = 66913566720;
but the smallest number with 1215 divisors is 3073593600 = 2^8 * 3^4 * 5^2 * 7^2 * 11^2, the smallest number with 1216 divisors is 35424829440 = 2^18 * 3^3 * 5 * 7 * 11 * 13;
so both 1215 and 1216 are exceptional, so 1215 is a term.

Cf. A005179, A037019, A072066.

isA331625(n) = A037019(n+1) > A005179(n+1) && A037019(n) > A005179(n) \\ See A037019 and A005179 for their programs. Warning: this is extremely inefficient

More terms from Jinyuan Wang, Jan 21 2025

cp's OEIS Frontend

A167755 Match the multisets in A175020 with those in A037016 then merge the two sequences.

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A125106 Enumeration of partitions by binary representation: each 1 is a part; the part size is 1 more than the number of 0's in the rest of the number.

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A037013 Reading binary expansion from right to left, run lengths strictly decrease.

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A037015 Numbers n with property that, reading binary expansion of n from right to left, run lengths strictly increase.

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A037014 Numbers n with property that reading binary expansion from right to left (from least significant to most significant), run lengths do not increase.

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A335835 Sort the run lengths in binary expansion of n in descending order (with multiplicities).

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PARI

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A331625 Numbers k such that both k and k+1 are exceptional (A072066).

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PARI

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