A062289 -id:A062289 - cp's OEIS frontend

A007461 Shifts left under AND-convolution with itself.

1, 1, 2, 1, 2, 4, 0, 5, 2, 4, 0, 10, 0, 12, 4, 13, 6, 12, 0, 18, 12, 20, 20, 36, 20, 36, 16, 44, 32, 60, 40, 73, 50, 56, 40, 58, 44, 52, 60, 84, 36, 112, 88, 108, 136, 132, 152, 178, 136, 232, 108, 260, 244, 256, 304, 288

Offset: 0

N. J. A. Sloane

a(A000225(n)) mod 2 = 1, a(A062289(n)) mod 2 = 0. [Reinhard Zumkeller, Apr 02 2012]

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to arXiv version]
M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to Lin. Alg. Applic. version together with omitted figures]

Cf. A007460, A199770.

import Data.Bits ((.&.))
a007461 n = a007461_list !! n
a007461_list = 1 : f [1,1] where
   f xs = x : f (x:xs) where
     x = sum $ zipWith (.&.) xs $ tail $ reverse xs :: Integer
-- Reinhard Zumkeller, Apr 02 2012

a:= proc(n) option remember; `if`(n=0, 1, add(
      Bits[And](a(i), a(n-1-i)), i=0..n-1))
    end:
seq(a(n), n=0..80);  # Alois P. Heinz, Jun 16 2018

a[0]=1; a[1]=1; a[n_] := a[n] = Sum[BitAnd[a[k], a[n-k-1]], {k, 0, n-1}]; Table[a[n], {n, 0, 60}] (* Jean-François Alcover, Sep 07 2012 *)

A326781 No position of a 1 in the reversed binary expansion of n is a power of 2.

Original entry on oeis.org

0, 4, 16, 20, 32, 36, 48, 52, 64, 68, 80, 84, 96, 100, 112, 116, 256, 260, 272, 276, 288, 292, 304, 308, 320, 324, 336, 340, 352, 356, 368, 372, 512, 516, 528, 532, 544, 548, 560, 564, 576, 580, 592, 596, 608, 612, 624, 628, 768, 772, 784, 788, 800, 804, 816

Offset: 1

Gus Wiseman, Jul 25 2019

Also BII-numbers (see A326031) of set-systems with no singleton edges. For example, the sequence of such set-systems together with their BII-numbers begins:
{}
{{1,2}}
{{1,3}}
{{1,2},{1,3}}
{{2,3}}
{{1,2},{2,3}}
{{1,3},{2,3}}
{{1,2},{1,3},{2,3}}
{{1,2,3}}
{{1,2},{1,2,3}}
{{1,3},{1,2,3}}
{{1,2},{1,3},{1,2,3}}
{{2,3},{1,2,3}}
{{1,2},{2,3},{1,2,3}}
{{1,3},{2,3},{1,2,3}}
{{1,2},{1,3},{2,3},{1,2,3}}
{{1,4}}
{{1,2},{1,4}}
{{1,3},{1,4}}
{{1,2},{1,3},{1,4}}

			The binary indices of n are row n of A048793. The sequence of terms together with their binary indices begins:
    0: {}
    4: {3}
   16: {5}
   20: {3,5}
   32: {6}
   36: {3,6}
   48: {5,6}
   52: {3,5,6}
   64: {7}
   68: {3,7}
   80: {5,7}
   84: {3,5,7}
   96: {6,7}
  100: {3,6,7}
  112: {5,6,7}
  116: {3,5,6,7}
  256: {9}
  260: {3,9}
  272: {5,9}
  276: {3,5,9}

Cf. A000120, A029931, A048793, A062289, A070939, A326031, A326782, A326788.

bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
Select[Range[100],!MemberQ[Length/@bpe/@bpe[#],1]&]

Conjectures from Colin Barker, Jul 27 2019: (Start)
G.f.: 4*x^2*(1 + 3*x + x^2 + 3*x^3 + x^4 + 3*x^5 + x^6 + 3*x^7 + x^8 + 3*x^9 + x^10 + 3*x^11 + x^12 + 3*x^13 + x^14 + 35*x^15) / ((1 - x)^2*(1 + x)*(1 + x^2)*(1 + x^4)*(1 + x^8)).
a(n) = a(n-1) + a(n-16) - a(n-17) for n>17.
(End)

A382413 Numbers with at least one zero in their base-7 representation.

Original entry on oeis.org

0, 7, 14, 21, 28, 35, 42, 49, 50, 51, 52, 53, 54, 55, 56, 63, 70, 77, 84, 91, 98, 99, 100, 101, 102, 103, 104, 105, 112, 119, 126, 133, 140, 147, 148, 149, 150, 151, 152, 153, 154, 161, 168, 175, 182, 189, 196, 197, 198, 199, 200, 201, 202, 203, 210, 217, 224, 231, 238

Offset: 1

Paolo Xausa, Mar 24 2025

Paolo Xausa, Table of n, a(n) for n = 1..10000

Cf. analogous sequences in other bases: A062289 (base 2), A081605 (base 3), A196032 (base 4), A382415 (base 5), A382416 (base 6), A382417 (base 8), A382418 (base 9), A011540 (base 10).

Cf. A007093, A043393, A382412 (complement).

Select[Range[0, 250], DigitCount[#, 7, 0] > 0 &]

A111066 Numbers with digits 1 and 2 and at least one of each.

Original entry on oeis.org

12, 21, 112, 121, 122, 211, 212, 221, 1112, 1121, 1122, 1211, 1212, 1221, 1222, 2111, 2112, 2121, 2122, 2211, 2212, 2221, 11112, 11121, 11122, 11211, 11212, 11221, 11222, 12111, 12112, 12121, 12122, 12211, 12212, 12221, 12222, 21111, 21112, 21121, 21122, 21211

Offset: 1

Alexandre Wajnberg & Youri Mora, Oct 08 2005

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Equals A007931 minus A000042 and A002276. Supersequence of A214218.

Cf. A043494, A037415, A062289, A000225.

FromDigits /@ Select[ IntegerDigits[ Range[210], 3], Union[ # ] == {1, 2} &] (* Robert G. Wilson v, Oct 09 2005 *)
Union[FromDigits/@Select[Flatten[Table[Tuples[{1,2},n],{n,2,5}],1], Union[#] == {1,2}&]] (* Harvey P. Dale, Sep 05 2013 *)

from itertools import count, islice
def agen():
    for i in count(1):
        s = bin(i+1)[3:].replace('1', '2').replace('0', '1')
        if 0 < s.count('1') < len(s):
            yield int(s)
print(list(islice(agen(), 42))) # Michael S. Branicky, Dec 21 2021

More terms from Robert G. Wilson v, Oct 09 2005

Crossrefs from Charles R Greathouse IV, Aug 03 2010

A382415 Numbers with at least one zero in their base-5 representation.

Original entry on oeis.org

0, 5, 10, 15, 20, 25, 26, 27, 28, 29, 30, 35, 40, 45, 50, 51, 52, 53, 54, 55, 60, 65, 70, 75, 76, 77, 78, 79, 80, 85, 90, 95, 100, 101, 102, 103, 104, 105, 110, 115, 120, 125, 126, 127, 128, 129, 130, 131, 132, 133, 134, 135, 136, 137, 138, 139, 140, 141, 142, 143, 144, 145

Offset: 1

Paolo Xausa, Mar 25 2025

Paolo Xausa, Table of n, a(n) for n = 1..10000

Cf. analogous sequences in other bases: A062289 (base 2), A081605 (base 3), A196032 (base 4), A382416 (base 6), A382413 (base 7), A382417 (base 8), A382418 (base 9), A011540 (base 10).

Cf. A007091, A023721 (complement), A023722.

Select[Range[0, 150], DigitCount[#, 5, 0] > 0 &]

A382416 Numbers with at least one zero in their base-6 representation.

Original entry on oeis.org

0, 6, 12, 18, 24, 30, 36, 37, 38, 39, 40, 41, 42, 48, 54, 60, 66, 72, 73, 74, 75, 76, 77, 78, 84, 90, 96, 102, 108, 109, 110, 111, 112, 113, 114, 120, 126, 132, 138, 144, 145, 146, 147, 148, 149, 150, 156, 162, 168, 174, 180, 181, 182, 183, 184, 185, 186, 192, 198

Offset: 1

Paolo Xausa, Mar 25 2025

Paolo Xausa, Table of n, a(n) for n = 1..10000

Cf. analogous sequences in other bases: A062289 (base 2), A081605 (base 3), A196032 (base 4), A382415 (base 5), A382413 (base 7), A382417 (base 8), A382418 (base 9), A011540 (base 10).

Cf. A007092, A043369, A248910 (complement).

Select[Range[0, 200], DigitCount[#, 6, 0] > 0 &]

A382417 Numbers with at least one zero in their base-8 representation.

Original entry on oeis.org

0, 8, 16, 24, 32, 40, 48, 56, 64, 65, 66, 67, 68, 69, 70, 71, 72, 80, 88, 96, 104, 112, 120, 128, 129, 130, 131, 132, 133, 134, 135, 136, 144, 152, 160, 168, 176, 184, 192, 193, 194, 195, 196, 197, 198, 199, 200, 208, 216, 224, 232, 240, 248, 256, 257, 258, 259, 260

Offset: 1

Paolo Xausa, Mar 25 2025

Paolo Xausa, Table of n, a(n) for n = 1..10000

Cf. analogous sequences in other bases: A062289 (base 2), A081605 (base 3), A196032 (base 4), A382415 (base 5), A382416 (base 6), A382413 (base 7), A382418 (base 9), A011540 (base 10).

Cf. A007094, A043421, A255805 (complement).

Select[Range[0, 300], DigitCount[#, 8, 0] > 0 &]

A382418 Numbers with at least one zero in their base-9 representation.

Original entry on oeis.org

0, 9, 18, 27, 36, 45, 54, 63, 72, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 99, 108, 117, 126, 135, 144, 153, 162, 163, 164, 165, 166, 167, 168, 169, 170, 171, 180, 189, 198, 207, 216, 225, 234, 243, 244, 245, 246, 247, 248, 249, 250, 251, 252, 261, 270, 279, 288, 297

Offset: 1

Paolo Xausa, Mar 25 2025

Paolo Xausa, Table of n, a(n) for n = 1..10000

Cf. analogous sequences in other bases: A062289 (base 2), A081605 (base 3), A196032 (base 4), A382415 (base 5), A382416 (base 6), A382413 (base 7), A382417 (base 8), A011540 (base 10).

Cf. A007095, A043453, A255808 (complement).

Select[Range[0, 300], DigitCount[#, 9, 0] > 0 &]

A132782 Numbers with at least one occurrence of '1010' in binary representation.

Original entry on oeis.org

10, 20, 21, 26, 40, 41, 42, 43, 52, 53, 58, 74, 80, 81, 82, 83, 84, 85, 86, 87, 90, 104, 105, 106, 107, 116, 117, 122, 138, 148, 149, 154, 160, 161, 162, 163, 164, 165, 166, 167, 168, 169, 170, 171, 172, 173, 174, 175, 180, 181, 186, 202, 208, 209, 210, 211, 212

Offset: 1

Reinhard Zumkeller, Aug 30 2007

Complement of A132781; subsequence of A062289; A000975 is a subsequence apart of the initial 4 terms.

q:= n-> verify([0, 1, 0, 1], Bits[Split](n), 'sublist'):
select(q, [$0..300])[];  # Alois P. Heinz, Oct 22 2021

Select[Range[250],SequenceCount[IntegerDigits[#,2],{1,0,1,0}]>0&] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Jul 08 2017 *)

A138836 Non-Mersenne numbers A001348.

Original entry on oeis.org

1, 2, 4, 5, 6, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70

Offset: 1

Omar E. Pol, Apr 05 2008

nonn

Numbers that are not in A001348.

a(1) to a(2042) equals A133398, then a(2043)=2048 <> A133398(2043)=2047.

Chai Wah Wu, Algorithms for complementary sequences, arXiv:2409.05844 [math.NT], 2024.

Cf. A001348, A001690, A062289, A133398, A138837, A138888, A138890, A138891. A133398.

from sympy import primepi, prime
def A138836(n): return n+(k:=int(primepi((n).bit_length())-1))+int(n+k+1>=1<1 else 1 # Chai Wah Wu, Sep 10 2024

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A007461 Shifts left under AND-convolution with itself.

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A326781 No position of a 1 in the reversed binary expansion of n is a power of 2.

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A382413 Numbers with at least one zero in their base-7 representation.

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A111066 Numbers with digits 1 and 2 and at least one of each.

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A382415 Numbers with at least one zero in their base-5 representation.

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A382416 Numbers with at least one zero in their base-6 representation.

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A382417 Numbers with at least one zero in their base-8 representation.

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A382418 Numbers with at least one zero in their base-9 representation.

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A132782 Numbers with at least one occurrence of '1010' in binary representation.

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