A230300 -id:A230300 - cp's OEIS frontend

A092391 a(n) = n + wt(n), where wt(n) = A000120(n) = binary weight of n.

0, 2, 3, 5, 5, 7, 8, 10, 9, 11, 12, 14, 14, 16, 17, 19, 17, 19, 20, 22, 22, 24, 25, 27, 26, 28, 29, 31, 31, 33, 34, 36, 33, 35, 36, 38, 38, 40, 41, 43, 42, 44, 45, 47, 47, 49, 50, 52, 50, 52, 53, 55, 55, 57, 58, 60, 59, 61, 62, 64, 64, 66, 67, 69, 65, 67, 68, 70, 70, 72, 73, 75

Offset: 0

Reinhard Zumkeller, May 08 2004

Reinhard Zumkeller and Donovan Johnson, Table of n, a(n) for n = 0..10000 (terms up to a(1023) from Reinhard Zumkeller)
Max A. Alekseyev and N. J. A. Sloane, On Kaprekar's Junction Numbers, arXiv:2112.14365, 2021; Journal of Combinatorics and Number Theory 12:3 (2022), 115-155.
Index entries for Colombian or self numbers and related sequences

A010061 gives the numbers not occurring in this sequence. A228082 gives the terms of this sequence sorted into ascending order, with duplicates removed. A228085(n) gives the number of times n occurs in this sequence.
Cf. A000120, A228083, A228086, A228087, A228091, A011371, A230058, A230092, A062028.
Cf. A228088, A230091, A227915, A230300.

a092391 n = n + a000120 n  -- Reinhard Zumkeller, May 13 2012

Table[n + Total[IntegerDigits[n, 2]], {n, 0, 100}] (* Jean-François Alcover, Sep 03 2013 *)

A092391(n)=n+hammingweight(n) \\ M. F. Hasler, Oct 05 2013

def a(n): return n + bin(n).count("1")
print([a(n) for n in range(72)]) # Michael S. Branicky, May 26 2022

a(n) = n + A000120(n).
A010062(n+1) = a(A010062(n)).
G.f.: (1/(1 - x))*Sum_{k>=0} (2^k + 1)*x^(2^k)/(1 + x^(2^k)). - Ilya Gutkovskiy, Jul 23 2017

A230301 Positive numbers not of the form m + wt(m-1), m >= 1.

Original entry on oeis.org

2, 5, 7, 14, 16, 19, 22, 24, 31, 33, 38, 40, 47, 49, 52, 55, 57, 64, 72, 79, 81, 84, 87, 89, 96, 98, 103, 105, 112, 114, 117, 120, 122, 129, 131, 134, 136, 143, 145, 148, 151, 153, 160, 162, 167, 169, 176, 178, 181, 184, 186, 193, 201, 208, 210, 213, 216, 218, 225, 227, 232, 234, 241, 243, 246, 249, 251, 271, 273, 276

Offset: 1

N. J. A. Sloane, Oct 23 2013

wt(m) = A000120(m).

These are numbers k such that A228085(2^k) = A228085(k-1) = 0, or numbers k such that 2^k is a binary self number (A010061). - Amiram Eldar, Feb 23 2021

Index entries for Colombian or self numbers and related sequences

Cf. A000120, A010061, A228085, A230300, A232491.

a(n) = A010061(n) + 1.

A374348 a(n) = k where wt(k) = n and k + wt(k) = a power of two, where wt(n) = A000120(n) = binary weight of n.

Original entry on oeis.org

1, 6, 13, 60, 59, 250, 505, 2040, 1015, 4086, 8181, 32756, 32755, 131058, 262129, 1048560, 262127, 1048558, 2097133, 8388588, 8388587, 33554410, 67108841, 268435432, 134217703, 536870886, 1073741797, 4294967268, 4294967267, 17179869154, 34359738337, 137438953440

Offset: 1

Steven Reyes, Jul 05 2024

k is uniquely determined by finding the power of two for which k = 2^x - n has wt(k) = n.

Terms are not always increasing, since the number of 0 bits in n-1 reduces k.

			For n = 4, 60 in binary is 111100, which has sum of digits of 4, and 60 + 4 = 64, a power of two.
For n = 5, 59 in binary is 111011, which has sum of digits of 5, and 59 + 5 = 64.

Alois P. Heinz, Table of n, a(n) for n = 1..1000

Cf. A000079, A000120, A092391, A129195, A230300.

a:= n-> 2^(n+add(i, i=Bits[Split](n-1)))-n:
seq(a(n), n=1..32);  # Alois P. Heinz, Jul 05 2024

def a(n):
  return (1 << (n + (n-1).bit_count())) - n

a(n) = 2^A230300(n) - n.
a(n) = 2^(n + A000120(n-1)) - n.
a(n) = 2 * A129195(n-1) - n.
a(n) == n (mod 2).

cp's OEIS Frontend

A092391 a(n) = n + wt(n), where wt(n) = A000120(n) = binary weight of n.

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A230301 Positive numbers not of the form m + wt(m-1), m >= 1.

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A374348 a(n) = k where wt(k) = n and k + wt(k) = a power of two, where wt(n) = A000120(n) = binary weight of n.

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