A350601 -id:A350601 - cp's OEIS frontend

A228082 Numbers that are of the form k + sum of binary digits of k for some nonnegative integer k.

0, 2, 3, 5, 7, 8, 9, 10, 11, 12, 14, 16, 17, 19, 20, 22, 24, 25, 26, 27, 28, 29, 31, 33, 34, 35, 36, 38, 40, 41, 42, 43, 44, 45, 47, 49, 50, 52, 53, 55, 57, 58, 59, 60, 61, 62, 64, 65, 66, 67, 68, 69, 70, 72, 73, 74, 75, 76, 77, 79, 81, 82, 84, 85, 87, 89, 90

Offset: 1

Antti Karttunen, Aug 09 2013

Complement of A010061.
Obtained when A092391 is sorted and duplicates are removed.
The asymptotic density of this sequence is 1 - (1/8) * (Sum_{n>=1} 1/2^a(n))^2 = 1 - A242403 = 0.747339... - Amiram Eldar, Nov 28 2020

Steven R. Finch, Mathematical Constants, Cambridge, 2003, Section 2.24, pp. 179-180.
József Sándor and Borislav Crstici, Handbook of Number theory II, Kluwer Academic Publishers, 2004, Chapter 4, p. 384-386.
G. Troi and U. Zannier, Note on the density constant in the distribution of self-numbers, Bolletino U. M. I. (7) 9-A (1995), 143-148.

Antti Karttunen, Table of n, a(n) for n = 1..8192
G. Troi and U. Zannier, Note on the density constant in the distribution of self-numbers. II, Bollettino dell'Unione Matematica Italiana 2-A (1999), 397-399.
Index entries for Colombian or self numbers and related sequences

Numbers that occur to the right of the leftmost column of A228083. Positions of nonzeros in A228085. A superset of A228088.
Cf. also A227643, A005187, A230091, A230092, A227915, A242403.
The even terms are the first row of A350601.

a228082 n = a228082_list !! (n-1)
a228082_list = 0 : filter ((> 0) . a228085) [1..]
-- Reinhard Zumkeller, Oct 13 2013

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

A230624 Numbers k with property that for every base b >= 2, there is a number m such that m+s(m) = k, where s(m) = sum of digits in the base-b expansion of m.

Original entry on oeis.org

0, 2, 10, 14, 22, 38, 62, 94, 158, 206, 318, 382, 478, 606, 766, 958, 1022, 1534, 1662, 1726, 1790, 1918, 1982, 2238, 2622, 2686, 3006, 3262, 3582, 3966, 4734, 5118, 5374, 5758, 5886, 6782, 8830, 9342, 9470, 9598, 10878, 12926, 13182, 13438, 14718, 18686, 22526

Offset: 1

N. J. A. Sloane, Oct 27 2013

If k is a positive term then k is even (or else k has no generator in base k+1) but not a multiple of 4 (or else k has no generator in base k/2). - David Applegate, Jan 09 2022. See A349821 and A350607 for the k/2 and (k-2)/4 sequences.
It is not known if this sequence is infinite.
The eight terms 10 through 206 are all twice primes (cf. A349820).

			10 is a member because in base 2, 7=111, 7+3=10; in base 3, 7=21, 7+3=10; in base 4, 8=20, 8+2=10; in base 5, 7=12, 7+3=10; and in bases b >= 6, 5+5=10.

David Applegate, Table of n, a(n) for n = 1..547, terms < 10^9 (first 90 terms from Lars Blomberg)
David Applegate, Two graphs to accompany Comments (see next link)
David Applegate and N. J. A. Sloane, Comments on A230624, Numbers that are generated in every base
Santanu Bandyopadhyay, Self-Number, Indian Institute of Technology Bombay (Mumbai, India, 2020).
Santanu Bandyopadhyay, Self-Number, Indian Institute of Technology Bombay (Mumbai, India, 2020). [Local copy]
Cai, Tianxin, On k-self-numbers and universal generated numbers, Fibonacci Quart. 34 (1996), no. 2, 144--146. MR1386983 (97c:11008)
Index entries for Colombian or self numbers and related sequences

Cf. A003052, A349820, A349821, A349822, A350607.
For first differences see A349823.
This is the limiting row of A350601.

More terms from Lars Blomberg, Oct 12 2015

More terms from David Applegate, Jan 02 2022

A349831 Even numbers in the intersection of A228082 and A349829.

Original entry on oeis.org

0, 2, 10, 12, 14, 16, 22, 24, 26, 28, 34, 36, 38, 40, 44, 50, 58, 60, 62, 66, 68, 70, 72, 74, 76, 82, 84, 92, 94, 96, 98, 106, 108, 110, 114, 118, 120, 122, 126, 132, 134, 136, 140, 146, 154, 156, 158, 162, 164, 170, 174, 176, 178, 186, 188, 190, 196, 198, 202, 204, 206, 210, 214, 216, 218, 222

Offset: 1

N. J. A. Sloane, Jan 07 2022

Even numbers that are "generated" (in Kaprekar's sense) in both bases 2 and 4.

Cf. A003052, A228082, A349829, A349830, A349832.
A230624 is a subsequence.
A row of A350601.

A349832 Even numbers that are "generated" (in Kaprekar's sense) in all three bases 2, 4, and 6.

Original entry on oeis.org

0, 2, 10, 14, 16, 22, 24, 28, 34, 36, 38, 44, 50, 58, 60, 62, 66, 68, 72, 74, 76, 82, 84, 92, 94, 96, 98, 106, 108, 110, 118, 120, 122, 126, 132, 134, 136, 140, 146, 154, 156, 158, 162, 164, 170, 176, 178, 186, 196, 198, 202, 206, 210, 214, 216, 222, 228, 234, 238, 244, 246, 252, 256, 258, 260

Offset: 1

N. J. A. Sloane, Jan 07 2022

Using Max Alekseyev's PARI "Gen" program (see A010061), we run
vector(500,k,length(Gen(k,2))),
vector(500,k,length(Gen(k,4))), and
vector(500,k,length(Gen(k,6)))
to find the numbers that are generated in bases 2, 4, and 6, and then take the even numbers that are common to all three lists.

Cf. A003052, A010061, A228082, A349829, A349830, A349831, A349833.
A230624 is a subsequence.
A row of A350601.

A349833 Even numbers that are "generated" (in Kaprekar's sense) in all four bases 2, 4, 6, and 8.

Original entry on oeis.org

0, 2, 10, 14, 22, 24, 28, 36, 38, 44, 50, 58, 60, 62, 66, 68, 74, 76, 82, 84, 92, 94, 96, 98, 106, 110, 118, 120, 122, 132, 134, 136, 140, 154, 156, 158, 162, 170, 176, 178, 186, 196, 198, 206, 210, 214, 216, 222, 228, 234, 244, 246, 252, 258, 260, 262, 264, 268, 274, 284, 286

Offset: 1

N. J. A. Sloane, Jan 07 2022

Using Max Alekseyev's PARI "Gen" program (see A010061), we run
vector(500,k,length(Gen(k,2))),
vector(500,k,length(Gen(k,4))),
vector(500,k,length(Gen(k,6))),
vector(500,k,length(Gen(k,8))),
to find the numbers that are generated in bases 2, 4, 6, and 8, and then take the even numbers that are common to all four lists.

N. J. A. Sloane, Table of n, a(n) for n = 1..2275

Cf. A003052, A010061, A228082, A349829, A349830, A349831, A349832.
A230624 is a subsequence.
A row of A350601.

cp's OEIS Frontend

A228082 Numbers that are of the form k + sum of binary digits of k for some nonnegative integer k.

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A230624 Numbers k with property that for every base b >= 2, there is a number m such that m+s(m) = k, where s(m) = sum of digits in the base-b expansion of m.

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A349831 Even numbers in the intersection of A228082 and A349829.

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A349832 Even numbers that are "generated" (in Kaprekar's sense) in all three bases 2, 4, and 6.

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A349833 Even numbers that are "generated" (in Kaprekar's sense) in all four bases 2, 4, 6, and 8.

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