A023689 -id:A023689 - cp's OEIS frontend

A067576 Array T(i,j) read by downward antidiagonals, where T(i,j) is the j-th term whose binary expansion has i 1's.

1, 2, 3, 4, 5, 7, 8, 6, 11, 15, 16, 9, 13, 23, 31, 32, 10, 14, 27, 47, 63, 64, 12, 19, 29, 55, 95, 127, 128, 17, 21, 30, 59, 111, 191, 255, 256, 18, 22, 39, 61, 119, 223, 383, 511, 512, 20, 25, 43, 62, 123, 239, 447, 767, 1023, 1024, 24, 26, 45, 79, 125, 247, 479, 895, 1535, 2047

Offset: 1

Robert G. Wilson v, Jan 30 2002

This is a permutation of the positive integers; the inverse permutation is A356419. - Jianing Song, Aug 06 2022

			Array begins:
        j=1  j=2  j=3  j=4  j=5  j=6
  i=1:    1,   2,   4,   8,  16,  32, ...
  i=2:    3,   5,   6,   9,  10,  12, ...
  i=3:    7,  11,  13,  14,  19,  21, ...
  i=4:   15,  23,  27,  29,  30,  39, ...
  i=5:   31,  47,  55,  59,  61,  62, ...
  i=6:   63,  95, 111, 119, 123, 125, ...

Ivan Neretin, Table of n, a(n) for n = 1..8001 (126 antidiagonals)
Index entries for sequences that are permutations of the natural numbers

Cf. A000120, A356419.
T(n,n) gives A036563(n+1).
The antidiagonals are read in the opposite direction from those in A066884.
Antidiagonal sums give A361074.
Cf. A000079, A018900, A014311, A014312, A014313, A023688, A023689, A023690, A023691.

a = {}; Do[ a = Append[a, Last[ Take[ Select[ Range[2^13], Count[ IntegerDigits[ #, 2], 1] == j & ], i - j]]], {i, 2, 12}, {j, 1, i - 1} ]; a

A066884 Square array read by upward antidiagonals where the n-th row contains the positive integers with n binary 1's.

Original entry on oeis.org

1, 3, 2, 7, 5, 4, 15, 11, 6, 8, 31, 23, 13, 9, 16, 63, 47, 27, 14, 10, 32, 127, 95, 55, 29, 19, 12, 64, 255, 191, 111, 59, 30, 21, 17, 128, 511, 383, 223, 119, 61, 39, 22, 18, 256, 1023, 767, 447, 239, 123, 62, 43, 25, 20, 512, 2047, 1535, 895, 479, 247, 125, 79, 45, 26, 24, 1024

Offset: 1

Jared Benjamin Ricks (jaredricks(AT)yahoo.com), Jan 21 2002

This is a permutation of the positive integers; the inverse permutation is A067587.

			Column: 1   2   3   4   5   6
-----------------------------
Row 1:| 1   2   4   8  16  32
Row 2:| 3   5   6   9  10  12
Row 3:| 7  11  13  14  19  21
Row 4:|15  23  27  29  30  39
Row 5:|31  47  55  59  61  62
Row 6:|63  95 111 119 123 125

Ivan Neretin, Table of n, a(n) for n = 1..8001 (126 antidiagonals)
Vladimir Dobric, M. Skyers, and L. J. Stanley, Polynomial Time Computable Triangular Arrays For Almost Sure Convergence, arXiv preprint arXiv:1603.04896 [math.PR], 2016. [Shows that this sequence is in P-TIME]
Index entries for sequences that are permutations of the natural numbers

Cf. A000120, A067587.
Selected rows: A000079 (1), A018900 (2), A014311 (3), A014312 (4), A014313 (5), A023688 (6), A023689 (7), A023690 (8), A023691 (9), A038461 (10), A038462 (11), A038463 (12). For decimal analogs, see A011557 and A038444-A038452.
Selected columns: A000225 (1), A055010 (2).
Selected diagonals: A036563 (main), A000918 (1st upper), A153894 (2nd upper). [Franklin T. Adams-Watters, Apr 22 2009]
Cf. A067576 (the same array read by downward antidiagonals).
Antidiagonal sums give A361074.

a = {}; Do[ a = Append[a, Last[ Take[ Take[ Select[ Range[2^12], Count[ IntegerDigits[ #, 2], 1] == j - i + 1 & ], j], i]]], {j, 1, 11}, {i, 1, j}]; a

Corrected and extended by Henry Bottomley, Jan 27 2002

A038461 Sums of 10 distinct powers of 2.

Original entry on oeis.org

1023, 1535, 1791, 1919, 1983, 2015, 2031, 2039, 2043, 2045, 2046, 2559, 2815, 2943, 3007, 3039, 3055, 3063, 3067, 3069, 3070, 3327, 3455, 3519, 3551, 3567, 3575, 3579, 3581, 3582, 3711, 3775, 3807, 3823, 3831, 3835, 3837, 3838, 3903

Offset: 1

Olivier Gérard

Ivan Neretin, Table of n, a(n) for n = 1..10000
Robert Baillie, Summing the curious series of Kempner and Irwin, arXiv:0806.4410 [math.CA], 2008-2015. See p. 18 for Mathematica code irwinSums.m.

Base 2 interpretation of A038452.

Cf. A000079, A018900, A014311, A014312, A014313, A023688, A023689, A023690, A023691 (Hamming weight = 1, 2, ..., 9).

Select[Range[4000], DigitCount[#, 2, 1] == 10 &] (* Amiram Eldar, Feb 14 2022 *)

isok(n) = hammingweight(n) == 10; \\ Michel Marcus, Feb 29 2016

from itertools import islice
def A038461_gen(): # generator of terms
    yield (n:=1023)
    while True: yield (n:=n^((a:=-n&n+1)|(a>>1)) if n&1 else ((n&~(b:=n+(a:=n&-n)))>>a.bit_length())^b)
A038461_list = list(islice(A038461_gen(),20)) # Chai Wah Wu, Mar 10 2025

Sum_{n>=1} 1/a(n) = 1.386312271262110321181505974797071257205562524228381227122302929089588534920... (calculated using Baillie's irwinSums.m, see Links). - Amiram Eldar, Feb 14 2022

Offset changed to 1 by Ivan Neretin, Feb 28 2016

A038462 Sums of 11 distinct powers of 2.

Original entry on oeis.org

2047, 3071, 3583, 3839, 3967, 4031, 4063, 4079, 4087, 4091, 4093, 4094, 5119, 5631, 5887, 6015, 6079, 6111, 6127, 6135, 6139, 6141, 6142, 6655, 6911, 7039, 7103, 7135, 7151, 7159, 7163, 7165, 7166, 7423, 7551, 7615, 7647, 7663, 7671

Offset: 1

Olivier Gérard

Ivan Neretin, Table of n, a(n) for n = 1..10000
Robert Baillie, Summing the curious series of Kempner and Irwin, arXiv:0806.4410 [math.CA], 2008-2015. See p. 18 for Mathematica code irwinSums.m.

Base 2 interpretation of A038453.

Cf. A000079, A018900, A014311, A014312, A014313, A023688, A023689, A023690, A023691, A038461 (Hamming weight = 1, 2, ..., 10).

Select[Range[8000], DigitCount[#, 2, 1] == 11 &] (* Amiram Eldar, Feb 14 2022 *)

from itertools import islice
def A038462_gen(): # generator of terms
    yield (n:=2047)
    while True: yield (n:=n^((a:=-n&n+1)|(a>>1)) if n&1 else ((n&~(b:=n+(a:=n&-n)))>>a.bit_length())^b)
A038462_list = list(islice(A038462_gen(),20)) # Chai Wah Wu, Mar 10 2025

Sum_{n>=1} 1/a(n) = 1.386300330514503033229968047555778179200262625510401687087371496738972082061... (calculated using Baillie's irwinSums.m, see Links). - Amiram Eldar, Feb 14 2022

Offset changed to 1 by Ivan Neretin, Feb 28 2016

A038463 Sums of 12 distinct powers of 2.

Original entry on oeis.org

4095, 6143, 7167, 7679, 7935, 8063, 8127, 8159, 8175, 8183, 8187, 8189, 8190, 10239, 11263, 11775, 12031, 12159, 12223, 12255, 12271, 12279, 12283, 12285, 12286, 13311, 13823, 14079, 14207, 14271, 14303, 14319, 14327, 14331, 14333

Offset: 1

Olivier Gérard

Ivan Neretin, Table of n, a(n) for n = 1..10000
Robert Baillie, Summing the curious series of Kempner and Irwin, arXiv:0806.4410 [math.CA], 2008-2015. See p. 18 for Mathematica code irwinSums.m.

Base 2 interpretation of A038454.

Cf. A000079, A018900, A014311, A014312, A014313, A023688, A023689, A023690, A023691, A038461, A038462 (Hamming weight = 1, 2, ..., 11).

Select[Range[15000], DigitCount[#, 2, 1] == 12 &] (* Amiram Eldar, Feb 14 2022 *)

from itertools import islice
def A038463_gen(): # generator of terms
    yield (n:=4095)
    while True: yield (n:=n^((a:=-n&n+1)|(a>>1)) if n&1 else ((n&~(b:=n+(a:=n&-n)))>>a.bit_length())^b)
A038463_list = list(islice(A038463_gen(),20)) # Chai Wah Wu, Mar 10 2025

Sum_{n>=1} 1/a(n) = 1.386296350824871649202152615241744383837323713474767661902780220440945591424... (calculated using Baillie's irwinSums.m, see Links). - Amiram Eldar, Feb 14 2022

Offset changed to 1 by Ivan Neretin, Feb 28 2016

A238015 Denominator of (2n+1)!8Bernoulli(2n,1/2).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 2, 4, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 2, 4, 1, 1, 1, 2, 1, 2, 2, 4, 1, 2, 2, 4, 2, 4, 4, 8, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 2, 4, 1, 1, 1, 2, 1, 2, 2, 4, 1, 2, 2, 4, 2, 4, 4, 8, 1, 1, 1, 2, 1

Offset: 0

Robert Israel, Feb 17 2014

It appears that a(n) is 1 for n in A095736, 2 for n in A014312, 4 for n in A014313, 8 for n in A023688, 16 for n in A023689, 32 for n in A023690, 64 for n in A023691. - Michel Marcus, Feb 18 2014

			For n=15, (2*15+1)!*8*Bernoulli(2*15,1/2) = -79147239268966167007717425917182573906640625/2 so a(15) = 2.

Robert Israel, Table of n, a(n) for n = 0..2000

Cf. A033473.

seq(denom((2*n+1)!*8*bernoulli(2*n,1/2)), n=0 .. 100);

Table[Denominator[(2 n + 1)! 8 BernoulliB[2 n, 1/2]], {n, 0, 200}] (* Vincenzo Librandi, Feb 18 2014 *)

A255568 Numbers in whose binary representation there are six 1-bits more than there are nonleading 0-bits.

Original entry on oeis.org

63, 191, 223, 239, 247, 251, 253, 254, 639, 703, 735, 751, 759, 763, 765, 766, 831, 863, 879, 887, 891, 893, 894, 927, 943, 951, 955, 957, 958, 975, 983, 987, 989, 990, 999, 1003, 1005, 1006, 1011, 1013, 1014, 1017, 1018, 1020, 2303, 2431, 2495, 2527, 2543, 2551, 2555

Offset: 1

Antti Karttunen, Mar 11 2015

Numbers for which A037861(n) = -6.

Numbers in whose binary representation (A007088) the number of 1-bits = 6 + number of (nonleading) 0 bits.

			63 ("111111" in binary) is included because there are 0 zero-bits and six 1-bits.
191 ("10111111" in binary) is included because there is 1 zero-bit and seven 1-bits, thus there are six 1-bits more than the number of 0-bits.

Antti Karttunen, Table of n, a(n) for n = 1..16303

Cf. A007088, A037861.

The intersection of A030130 and A023689 is a finite subsequence of this sequence.

is(n)=2*hammingweight(n)==6+#binary(n) \\ Charles R Greathouse IV, Dec 16 2015

for(b=3,6, for(n=2^(2*b-1),4^b-1, if(hammingweight(n)==3+b, print1(n", ")))) \\ Charles R Greathouse IV, Dec 16 2015

listBBitMembers(b)=if(b%2,return([])); my(u=List()); forvec(v=vector(3+b/2,i,[0,b-1]), listput(u, sum(i=1,#v,2^v[i])),2); Vec(u) \\ Charles R Greathouse IV, Dec 16 2015

use ntheory ":all"; my $bits = 0; for (1..1000) { my $o = hammingweight($); $bits++ if ($ & ($-1))==0; say if $o == 6+$bits-$o; } # _Dana Jacobsen, Dec 16 2015

A086772 Store the natural numbers in a triangular array such that values on each row have the same number of bits. Start a new row with the smallest number not yet recorded. a(n) represents the initial terms in the resulting array.

Original entry on oeis.org

0, 1, 3, 4, 7, 9, 15, 21, 24, 31, 41, 45, 63, 64, 72, 74, 83, 94, 127, 139, 140, 173, 197, 207, 234, 255, 268, 284, 288, 339, 349, 390, 426, 445, 467, 511, 522, 553, 569, 634, 689, 706, 734, 797, 838, 934, 950, 951, 1023, 1036, 1052, 1078, 1179, 1236

Offset: 0

Alford Arnold, Aug 03 2003

A067576 describes the sequences with a fixed number of binary bits using antidiagonals.

			The array begins:
   0
   1  2
   3  5  6
   4  8 16 32
   7 11 13 14 19
   9 10 12 17 18 20
  15 23 27 29 30 39 43
  ...
so the initial terms are 0 1 3 4 7 9 15 ...

Alois P. Heinz, Table of n, a(n) for n = 0..10000

Cf. A000079, A001477, A018900, A014311, A014312, A014313, A023688, A023689, A023690, A023691.

A086772aux := proc(n,k)
    option remember;
    local a,npr,kpr,fnd ;
    if n = 0 then
        return 0;
    end if;
    if k = 0 then
        for a from 1 do
            fnd := false;
            for npr from 1 to n-1 do
                for kpr from 0 to npr do
                    if procname(npr,kpr) = a then
                        fnd := true;
                        break;
                    end if;
                end do:
            end do:
            if not fnd then
                return a;
            end if;
        end do:
    else
        for a from 1 do
            if wt(a) = wt(procname(n,0)) then
                fnd := false;
                for npr from 1 to n-1 do
                    for kpr from 0 to npr do
                        if procname(npr,kpr) = a then
                            fnd := true;
                            break;
                        end if;
                    end do:
                end do:
                for kpr from 0 to k-1 do
                    if procname(n,kpr) = a then
                        fnd := true;
                        break;
                    end if;
                end do:
                if not fnd then
                    return a;
                end if;
            end if;
        end do:
    end if;
end proc:
A086772 := proc(n)
    A086772aux(n,0) ;
end proc: # R. J. Mathar, Sep 15 2012

A367680 Number of integer compositions x1+x2+...+xk of n such that each xj has exactly j bits set.

Original entry on oeis.org

1, 1, 1, 0, 2, 1, 1, 3, 2, 1, 2, 4, 2, 4, 6, 2, 4, 5, 10, 7, 10, 12, 8, 6, 11, 14, 16, 13, 16, 16, 14, 14, 30, 32, 19, 35, 28, 23, 27, 38, 36, 47, 44, 42, 55, 52, 51, 85, 88, 74, 84, 84, 72, 81, 102, 110, 122, 115, 108, 132, 137, 136, 179, 195, 164, 160, 181

Offset: 0

Arnauld Chevallier, Nov 26 2023

			There are 6 such compositions for n = 14:
  14 = 1 + 6 + 7 (1 + 110 + 111)
  14 = 2 + 5 + 7 (10 + 101 + 111)
  14 = 2 + 12 (10 + 1100)
  14 = 4 + 3 + 7 (100 + 11 + 111)
  14 = 4 + 10 (100 + 1010)
  14 = 8 + 6 (1000 + 110)
Therefore a(14) = 6.

Cf. A000079, A000120, A018900, A014311, A014313, A023688, A023689, A023690, A023691.

a(n) = my(nb=0); forpart(v=n, if (vecsort(apply(hammingweight, Vec(v))) == [1..#v], nb++)); nb; \\ Michel Marcus, Nov 28 2023

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A067576 Array T(i,j) read by downward antidiagonals, where T(i,j) is the j-th term whose binary expansion has i 1's.

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A066884 Square array read by upward antidiagonals where the n-th row contains the positive integers with n binary 1's.

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A038461 Sums of 10 distinct powers of 2.

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A038462 Sums of 11 distinct powers of 2.

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A038463 Sums of 12 distinct powers of 2.

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A238015 Denominator of (2*n+1)!*8*Bernoulli(2*n,1/2).

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A255568 Numbers in whose binary representation there are six 1-bits more than there are nonleading 0-bits.

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A238015 Denominator of (2n+1)!8Bernoulli(2n,1/2).