A047457 -id:A047457 - cp's OEIS frontend

A063787 a(2^k) = k + 1 and a(2^k + i) = 1 + a(i) for k >= 0 and 0 < i < 2^k.

1, 2, 2, 3, 2, 3, 3, 4, 2, 3, 3, 4, 3, 4, 4, 5, 2, 3, 3, 4, 3, 4, 4, 5, 3, 4, 4, 5, 4, 5, 5, 6, 2, 3, 3, 4, 3, 4, 4, 5, 3, 4, 4, 5, 4, 5, 5, 6, 3, 4, 4, 5, 4, 5, 5, 6, 4, 5, 5, 6, 5, 6, 6, 7, 2, 3, 3, 4, 3, 4, 4, 5, 3, 4, 4, 5, 4, 5, 5, 6, 3, 4, 4, 5, 4, 5, 5, 6, 4, 5, 5, 6, 5, 6, 6, 7, 3, 4, 4, 5, 4, 5, 5, 6, 4

Offset: 1

Reinhard Zumkeller, Aug 16 2001

nonn

Hamming weights of odd numbers. - Friedjof Tellkamp, Jan 11 2024

			k = 3: a(2^3) = a(8) = 4 = 3 + 1.
k = 3, i = 5: a(2^3 + 5) = a(13) = 3 = 1 + 2 = 1 + a(5).
From _Omar E. Pol_, Jun 12 2009: (Start)
Triangle begins:
  1;
  2,2;
  3,2,3,3;
  4,2,3,3,4,3,4,4;
  5,2,3,3,4,3,4,4,5,3,4,4,5,4,5,5;
  6,2,3,3,4,3,4,4,5,3,4,4,5,4,5,5,6,3,4,4,5,4,5,5,6,4,5,5,6,5,6,6;
  7,2,3,3,4,3,4,4,5,3,4,4,5,4,5,5,6,3,4,4,5,4,5,5,6,4,5,5,6,5,6,6,7,3,4,4,5,...
(End)

Michael Gilleland, Some Self-Similar Integer Sequences

Cf. A000079, A000120, A007814, A086799, A047457, A131136.

Cf. A330038 (partial sums).

Table[DigitCount[2 n - 1, 2, 1], {n, 1, 105}] (* Friedjof Tellkamp, Jan 11 2024 *)

a(n) = hammingweight(n-1) + 1; \\ Michel Marcus, Nov 23 2022

def a(n): return bin(n-1).count('1') + 1
print([a(n) for n in range(1, 106)]) # Michael S. Branicky, Dec 16 2021

a(n) = A000120(n-1) + 1.
a(n) = log(A131136)/log(2). - Stephen Crowley, Aug 25 2008
a(n) = A007814(n) + A000120(n). - Gary W. Adamson, Jun 04 2009
a(n) = A000120(A086799(n)). - Reinhard Zumkeller, Jul 31 2010
a(n) = A000120(A047457(n)-1) = A000120(A047457(n)+1). - Ilya Lopatin, Mar 16 2014
a(n) = A000120(2n-1). - Friedjof Tellkamp, Jan 11 2024

A386971 Numbers k >= 1 such that w(k-r) + ... + w(k-1) = w(k+1) + ... + w(k+r) for some r >= 1 where w(i) is the binary weight of i (A000120).

Original entry on oeis.org

3, 4, 7, 8, 11, 12, 14, 15, 16, 17, 19, 20, 23, 24, 27, 28, 29, 31, 32, 34, 35, 36, 39, 40, 43, 44, 46, 47, 48, 49, 51, 52, 55, 56, 58, 59, 60, 63, 64, 67, 68, 69, 71, 72, 75, 76, 78, 79, 80, 81, 83, 84, 87, 88, 91, 92, 93, 95, 96, 98, 99, 100, 103, 104, 107

Offset: 1

Ctibor O. Zizka, Aug 11 2025

r = 1 for k from A047457, k = 8*x - 5 or k = 8*x - 4, x >= 1.

r = 2 for k = 32*x - 18, (2* 7th row of A097586) or k = 32*x - 15, (10th row of A097586), x >= 1.

			For k = 7: A000120(4) + A000120(5) + A000120(6) = A000120(8) + A000120(9) + A000120(10), thus 7 is a term.

Ctibor O. Zizka, A386971 plot of k vs. r

Cf. A000120, A047457, A097586.

q[k_] := Module[{s = 0, r = 1}, While[r < k && (r == 1 || s != 0), s += (DigitSum[k-r, 2] - DigitSum[k+r, 2]); r++]; 1 < r <= k && s ==0]; Select[Range[120], q] (* Amiram Eldar, Aug 12 2025 *)

A322261 Square array T(n, k) (n >= 0, k >= 0) read by antidiagonals upwards: the lengths of runs in binary expansion of T(n, k) correspond to the lengths of runs in binary expansion of n followed by the lengths of runs in binary expansion of k.

Original entry on oeis.org

0, 1, 1, 2, 2, 2, 3, 5, 5, 3, 4, 6, 10, 4, 4, 5, 9, 13, 11, 11, 5, 6, 10, 18, 12, 20, 10, 6, 7, 13, 21, 19, 27, 21, 9, 7, 8, 14, 26, 20, 36, 26, 22, 8, 8, 9, 17, 29, 27, 43, 37, 25, 23, 23, 9, 10, 18, 34, 28, 52, 42, 38, 24, 40, 22, 10, 11, 21, 37, 35, 59, 53

Offset: 0

Rémy Sigrist, Dec 01 2018

The array T is associative.

			Array T(n, k) begins (in decimal):
  n\k|  0   1   2   3   4   5   6   7    8    9   10   11   12
  ---+--------------------------------------------------------
    0|  0   1   2   3   4   5   6   7    8    9   10   11   12
    1|  1   2   5   4  11  10   9   8   23   22   21   20   19
    2|  2   5  10  11  20  21  22  23   40   41   42   43   44
    3|  3   6  13  12  27  26  25  24   55   54   53   52   51
    4|  4   9  18  19  36  37  38  39   72   73   74   75   76
    5|  5  10  21  20  43  42  41  40   87   86   85   84   83
    6|  6  13  26  27  52  53  54  55  104  105  106  107  108
    7|  7  14  29  28  59  58  57  56  119  118  117  116  115
    8|  8  17  34  35  68  69  70  71  136  137  138  139  140
Array T(n, k) begins (in binary):
  n\k |     0      1      10      11      100      101      110      111      1000
  ----+---------------------------------------------------------------------------
     0|     0      1      10      11      100      101      110      111      1000
     1|     1     10     101     100     1011     1010     1001     1000     10111
    10|    10    101    1010    1011    10100    10101    10110    10111    101000
    11|    11    110    1101    1100    11011    11010    11001    11000    110111
   100|   100   1001   10010   10011   100100   100101   100110   100111   1001000
   101|   101   1010   10101   10100   101011   101010   101001   101000   1010111
   110|   110   1101   11010   11011   110100   110101   110110   110111   1101000
   111|   111   1110   11101   11100   111011   111010   111001   111000   1110111
  1000|  1000  10001  100010  100011  1000100  1000101  1000110  1000111  10001000

Index entries for sequences related to binary expansion of n

Cf. A004757, A005811, A010078, A020330, A042963, A047457, A047617, A101211, A163621.

torl(n) = my (r=[]); while (n, r = concat(valuation(n+(n%2),2), r); n \= 2^r[1];); r
fromrl(r) = my (v=0); for (i=1, #r, v = (v + (i%2))*2^r[i]-(i%2)); v
T(n,k) = fromrl(concat(torl(n), torl(k)))

T(n, 0) = T(0, n) = n.
T(n, 1) = A042963(n+1).
T(n, 2) = A047617(n+1).
T(n, 3) = A047457(n+1).
T(1, n) = A010078(n+1).
T(2, n) = A004757(n) for any n > 0.
A005811(T(n, k)) = A005811(n) + A005811(k).
T(2*n, k) = A163621(2*n, k) for any n > 0 and k > 0.
T(2*n, 2*n) = A020330(2*n) for any n > 0.

A372718 Triangular numbers that are 2 mod 4, halved.

Original entry on oeis.org

3, 5, 33, 39, 95, 105, 189, 203, 315, 333, 473, 495, 663, 689, 885, 915, 1139, 1173, 1425, 1463, 1743, 1785, 2093, 2139, 2475, 2525, 2889, 2943, 3335, 3393, 3813, 3875, 4323, 4389, 4865, 4935, 5439, 5513, 6045, 6123, 6683, 6765, 7353, 7439, 8055, 8145, 8789, 8883, 9555, 9653

Offset: 1

Tanya Khovanova and the PRIMES STEP senior group, May 11 2024

The sum of the first 2*a(n) numbers of any Fibonacci-like sequence equals its (a(n)+2)-nd term times the a(n)-th Lucas number.

			10 is a triangular number that has a remainder of 2 when divided by 4. Therefore, its half, 5, is in this sequence. Moreover, the sum of the first 5*2 Fibonacci numbers is 143 (not counting zero). This sum is a product of 13, which is the (5+2 = 7)-th term of the Fibonacci sequence times 11, which is the fifth Lucas number.

Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

Cf. A000032, A000045, A000217. A372048, A372049, A372050, A372051, A372070.

Select[Table[n (n + 1)/2, {n, 200}], Mod[#, 4] == 2 &]/2

a(n) = A000217(A047457(n))/2 = A372070(n)/2.

cp's OEIS Frontend

A063787 a(2^k) = k + 1 and a(2^k + i) = 1 + a(i) for k >= 0 and 0 < i < 2^k.

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A386971 Numbers k >= 1 such that w(k-r) + ... + w(k-1) = w(k+1) + ... + w(k+r) for some r >= 1 where w(i) is the binary weight of i (A000120).

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A322261 Square array T(n, k) (n >= 0, k >= 0) read by antidiagonals upwards: the lengths of runs in binary expansion of T(n, k) correspond to the lengths of runs in binary expansion of n followed by the lengths of runs in binary expansion of k.

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A372718 Triangular numbers that are 2 mod 4, halved.

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