A092391 -id:A092391 - cp's OEIS frontend

A227915 Numbers of the form k + wt(k) for exactly four distinct k, where wt(k) = A000120(k) is the binary weight of k.

4102, 12295, 20487, 28680, 36871, 45064, 53256, 61449, 69639, 77832, 86024, 94217, 102408, 110601, 118793, 126986, 135175, 143368, 151560, 159753, 167944, 176137, 184329, 192522, 200712, 208905, 217097, 225290, 233481, 241674, 249866, 258059, 266247, 274440, 282632, 290825, 299016, 307209, 315401, 323594, 331784, 339977

Offset: 1

Reinhard Zumkeller, Oct 13 2013

Numbers occurring exactly four times in A092391: A228085(a(n)) = 4. For the first number that appears k times, see A230303.

			a(1) = 4102, the four k with A092391(k) = 4102 being:
4091 = '111111111011', A000120(4091) = 11, 4091 + 11 = 4102;
4092 = '111111111100', A000120(4092) = 12, 4092 + 10 = 4102;
4099 = '1000000000011', A000120(4099) = 3, 4099 + 3 = 4102;
4100 = '1000000000100', A000120(4100) = 2, 4100 + 2 = 4102.

Donovan Johnson, Table of n, a(n) for n = 1..10000
Index entries for Colombian or self numbers and related sequences

Cf. A092391, A228085, A230092, A230091, A228088, A010061, A230093, A230303.

a227915 n = a227915_list !! (n-1)
a227915_list = filter ((== 4) . a228085) [1..]

A228086 a(n) is the least k which satisfies n = k + bitcount(k), or 0 if no such k exists. Here bitcount(k) (or wt(k), A000120) gives the number of 1's in binary representation of nonnegative integer k.

Original entry on oeis.org

0, 0, 1, 2, 0, 3, 0, 5, 6, 8, 7, 9, 10, 0, 11, 0, 13, 14, 0, 15, 18, 0, 19, 0, 21, 22, 24, 23, 25, 26, 0, 27, 0, 29, 30, 33, 31, 0, 35, 0, 37, 38, 40, 39, 41, 42, 0, 43, 0, 45, 46, 0, 47, 50, 0, 51, 0, 53, 54, 56, 55, 57, 58, 0, 59, 64, 61, 62, 66, 63, 67, 0

Offset: 0

Antti Karttunen, Aug 09 2013

nonn

A083058(n)+1 gives a lower bound for nonzero terms, n-1 an upper bound.

Antti Karttunen, Table of n, a(n) for n = 0..8192
Index entries for Colombian or self numbers and related sequences

Cf. A228087, A228085, A335599. A010061 gives the positions of zeros after a(0). The union of A010061 and A228088 gives the positions where a(n) = A228087(n).

Cf. also A213723, A227643.

a[n_] := Module[{k}, For[k = n - Floor[Log[2, n]] - 1, k < n, k++, If[n == k + DigitCount[k, 2, 1], Return[k]]]; 0];
a /@ Range[0, 1000]; (* Jean-François Alcover, Nov 28 2020 *)

(define (A228086 n) (if (zero? n) n (let loop ((k (+ (A083058 n) 1))) (cond ((> k n) 0) ((= n (A092391 k)) k) (else (loop (+ 1 k)))))))

A230091 Numbers of the form k + wt(k) for exactly two distinct k, where wt(k) = A000120(k) is the binary weight of k.

Original entry on oeis.org

5, 14, 17, 19, 22, 31, 33, 36, 38, 47, 50, 52, 55, 64, 67, 70, 79, 82, 84, 87, 96, 98, 101, 103, 112, 115, 117, 120, 131, 132, 143, 146, 148, 151, 160, 162, 165, 167, 176, 179, 181, 184, 193, 196, 199, 208, 211, 213, 216, 225, 227, 230, 232, 241, 244, 246, 249, 258, 260, 262, 271, 274, 276, 279, 288, 290, 293, 295

Offset: 1

N. J. A. Sloane, Oct 10 2013

The positions of entries equal to 2 in A228085, or numbers that appear exactly twice in A092391.

Numbers that can be expressed as the sum of distinct terms of the form 2^n+1, n=0,1,... in exactly two ways.

			5 = 3 + 2 = 4 + 1, so 5 is in this list.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Index entries for Colombian or self numbers and related sequences

Cf. A000120, A010061, A092391, A228088, A228085, A230058, A230092.

Cf. A227915.

a230091 n = a230091_list !! (n-1)
a230091_list = filter ((== 2) . a228085) [1..]
-- Reinhard Zumkeller, Oct 13 2013

# Maple code for A000120, A092391, A228085, A010061, A228088, A230091, A230092
with(LinearAlgebra):
read transforms;
wt := proc(n) local w, m, i; w := 0; m := n; while m > 0 do i := m mod 2; w := w+i; m := (m-i)/2; od; w; end: # A000120
M:=1000;
lis1:=Array(0..M);
lis2:=Array(0..M);
ctmax:=4;
for i from 0 to ctmax do ct[i]:=Array(0..M); od:
for n from 0 to M do
m:=n+wt(n);
lis1[n]:=m;
if (m <= M) then lis2[m]:=lis2[m]+1; fi;
od:
t1:=[seq(lis1[i],i=0..M)]; # A092391
t2:=[seq(lis2[i],i=0..M)]; # A228085
COMPl(t1); # A010061
for i from 1 to M do h:=lis2[i];
if h <= ctmax then ct[h]:=[op(ct[h]),i]; fi; od:
len:=nops(ct[0]); [seq(ct[0][i],i=1..len)]; # A010061 again
len:=nops(ct[1]); [seq(ct[1][i],i=1..len)]; # A228088
len:=nops(ct[2]); [seq(ct[2][i],i=1..len)]; # A230091
len:=nops(ct[3]); [seq(ct[3][i],i=1..len)]; # A230092

nt = 100; (* number of terms to produce *)
S[kmax_] := S[kmax] = Table[k + Total[IntegerDigits[k, 2]], {k, 0, kmax}] // Tally // Select[#, #[[2]] == 2&][[All, 1]]& // PadRight[#, nt]&;
S[nt];
S[kmax = 2 nt];
While[S[kmax] =!= S[kmax/2], kmax *= 2];
S[kmax] (* Jean-François Alcover, Mar 04 2023 *)

A233272 a(n) = n + 1 + number of nonleading zeros in binary representation of n (A080791).

Original entry on oeis.org

1, 2, 4, 4, 7, 7, 8, 8, 12, 12, 13, 13, 15, 15, 16, 16, 21, 21, 22, 22, 24, 24, 25, 25, 28, 28, 29, 29, 31, 31, 32, 32, 38, 38, 39, 39, 41, 41, 42, 42, 45, 45, 46, 46, 48, 48, 49, 49, 53, 53, 54, 54, 56, 56, 57, 57, 60, 60, 61, 61, 63, 63, 64, 64, 71, 71, 72

Offset: 0

Antti Karttunen, Dec 12 2013

From Antti Karttunen, Jan 30 2022: (Start)
Write n in binary: 1ab..xyz, then a(n) = (1+1ab..xy) + (1+1ab..x) + ... + (1+1ab) + (1+1a) + (1+1) + (1+0) + 1. This method was found by LODA miner, see the assembly program at C. Krause link.
Proof: Compare to a similar formula given for A011371, with a(n) = a(floor(n/2)) + floor(n/2) to the new formula for this sequence which is a(n) = 1 + a(floor(n/2)) + floor(n/2), for n > 0 and a(0) = 1. It is easy to see that the difference between these, a(n) - A011371(n) = 1+A070939(n), for n > 0. As A011371(n) = n minus (number of 1's in binary expansion of n), then a(n) = 1 + (number of digits in binary expansion of n) + (n minus number of 1's in binary expansion of n) = 1 + n + (number of nonleading 0's in binary expansion of n), which indeed is the definition of this sequence.
(End)

Antti Karttunen, Table of n, a(n) for n = 0..8192
Christian Krause, et al, A mined LODA assembly source for this sequence
Index entries for sequences related to binary expansion of n

Bisection: A233273 (A120511 + 1). Iterates: A233271, A216431.

Cf. A000120, A005187, A011371, A054429, A070939, A080791, A092391.

DigitCount[#, 2, 0] + # + 1 & [Range[0, 100]] (* Paolo Xausa, Mar 01 2024 *)

A233272(n) = { my(s=1); while(n, n>>=1; s+=(1+n)); (s); }; \\ (After a LODA-assembly program found by a miner) - Antti Karttunen, Jan 30 2022

(define (A233272 n) (+ 1 n (A080791 n)))
;; Alternatively:
(define (A233272 n) (if (zero? n) 1 (+ n (A000120 (A054429 n)))))

a(n) = n + A080791(n) + 1.
For all n>=1, a(n) = n + A000120(A054429(n)).
a(0) = 1; for n > 1, a(n) = 1 + floor(n/2) + a(floor(n/2)). - (Found by LODA miner, see comments) - Antti Karttunen, Jan 30 2022

A228087 a(n) = largest k which satisfies n = k + bitcount(k), or 0 if no such k exists. Here bitcount(k) (A000120) gives the number of 1's in binary representation of nonnegative integer k.

Original entry on oeis.org

0, 0, 1, 2, 0, 4, 0, 5, 6, 8, 7, 9, 10, 0, 12, 0, 13, 16, 0, 17, 18, 0, 20, 0, 21, 22, 24, 23, 25, 26, 0, 28, 0, 32, 30, 33, 34, 0, 36, 0, 37, 38, 40, 39, 41, 42, 0, 44, 0, 45, 48, 0, 49, 50, 0, 52, 0, 53, 54, 56, 55, 57, 58, 0, 60, 64, 61, 65, 66, 63, 68, 0

Offset: 0

Antti Karttunen, Aug 09 2013

nonn

A083058(n)+1 gives a lower bound for nonzero terms, n-1 an upper bound.

Antti Karttunen, Table of n, a(n) for n = 0..8192
Index entries for Colombian or self numbers and related sequences

Cf. A228086, A228085. A010061 gives the positions of zeros after a(0). The union of A010061 and A228088 gives the positions where a(n) = A228086(n).

Cf. also A213724, A227643.

(define (A228087 n) (let loop ((k n)) (cond ((<= k (A083058 n)) 0) ((= n (A092391 k)) k) (else (loop (- k 1))))))

A243441 Primes p such that p + A000120(p) is also a prime, where A000120 = sum of digits in base 2 = Hamming weight.

Original entry on oeis.org

2, 3, 5, 17, 43, 163, 277, 311, 347, 373, 461, 479, 571, 643, 673, 821, 853, 857, 881, 977, 983, 1013, 1093, 1103, 1117, 1181, 1223, 1297, 1427, 1433, 1439, 1481, 1523, 1607, 1613, 1621, 1823, 1861, 1871, 1873, 2003, 2083, 2281, 2333, 2393, 2417, 2467, 2549

Offset: 1

Anthony Sand, Jun 05 2014

			2 + digitsum(2,base=2) = 2 + digitsum(10) = 2 + 1 = 3, which is prime.
3 + digitsum(11) = 3 + 2 = 5.
5 + digitsum(101) = 5 + 2 = 7.
17 + digitsum(10001) = 17 + 2 = 19.
43 + digitsum(101011) = 43 + 4 = 47.

Anthony Sand, Table of n, a(n) for n = 1..1000

Cf. A000120, A092391 (n + A000120(n)), A048519 (analog for base 10).

Cf. A243442 (analog for p - A000120(p)).

Select[Prime@ Range@ 400, PrimeQ[# + Total@ IntegerDigits[#, 2]] &] (* Michael De Vlieger, Nov 06 2018 *)

lista(lim) = forprime(p=2,lim, if (isprime(p+hammingweight(p)), print1(p, ", "))); \\ Michel Marcus, Jun 10 2014

Name edited by M. F. Hasler, Nov 07 2018

A228083 Table of binary Self-numbers and their descendants; square array T(r,c), with row r>=1, column c>=1, read by antidiagonals.

Original entry on oeis.org

1, 2, 4, 3, 5, 6, 5, 7, 8, 13, 7, 10, 9, 16, 15, 10, 12, 11, 17, 19, 18, 12, 14, 14, 19, 22, 20, 21, 14, 17, 17, 22, 25, 22, 24, 23, 17, 19, 19, 25, 28, 25, 26, 27, 30, 19, 22, 22, 28, 31, 28, 29, 31, 34, 32, 22, 25, 25, 31, 36, 31, 33, 36, 36, 33, 37

Offset: 1

Antti Karttunen, Aug 09 2013

			The top-left corner of the square array:
   1,  2,  3,  5,  7, 10, 12, 14, ...
   4,  5,  7, 10, 12, 14, 17, 19, ...
   6,  8,  9, 11, 14, 17, 19, 22, ...
  13, 16, 17, 19, 22, 25, 28, 31, ...
  15, 19, 22, 25, 28, 31, 36, 38, ...
  18, 20, 22, 25, 28, 31, 36, 38, ...
  21, 24, 26, 29, 33, 35, 38, 41, ...
  23, 27, 31, 36, 38, 41, 44, 47, ...
  ...
The non-initial terms on each row are obtained by adding to the preceding term the number of 1-bits in its binary representation (A000120).

First column: A010061. First row: A010062. Transpose: A228084. See A151942 for decimal analog.

Cf. also A000120, A092391, A227643, A228082, A228085, A228086, A228087, A228088, A228091, A218254.

nmax0 = 100;
nmax := Length[col[1]];
col[1] = Table[n + DigitCount[n, 2, 1], {n, 0, nmax0}] // Complement[Range[Last[#]], #]&;
col[k_] := col[k] = col[k - 1] + DigitCount[col[k-1], 2, 1];
T[n_, k_] := col[k][[n]];
Table[T[n-k+1, k], {n, 1, nmax}, {k, n, 1, -1}] // Flatten (* Jean-François Alcover, Nov 28 2020 *)

T(r,1) are those numbers not of form n + sum of binary digits of n (binary Self numbers) = A010061(r);

T(r,c) = T(r,c-1) + sum of binary digits of T(r,c-1) = A092391(T(r,c-1)).

A230058 Numbers of the form k + wt(k) for at least two distinct k, where wt(k) = A000120(k) is the binary weight of k.

Original entry on oeis.org

5, 14, 17, 19, 22, 31, 33, 36, 38, 47, 50, 52, 55, 64, 67, 70, 79, 82, 84, 87, 96, 98, 101, 103, 112, 115, 117, 120, 129, 131, 132, 134, 143, 146, 148, 151, 160, 162, 165, 167, 176, 179, 181, 184, 193, 196, 199, 208, 211, 213, 216, 225, 227, 230, 232, 241, 244

Offset: 1

Matthew C. Russell, Oct 07 2013

The positions of entries greater than 1 in A228085, or numbers that appear multiple times in A092391.

Numbers that can be expressed as the sum of distinct terms of the form 2^n+1, n=0,1,... in multiple ways.

			5 = 3 + 2 = 4 + 1, so 5 is in this list.

Vincenzo Librandi and Donovan Johnson, Table of n, a(n) for n = 1..10000 (first 1000 terms from Vincenzo Librandi)
Index entries for Colombian or self numbers and related sequences

Cf. A000120, A010061, A228088, A230091, A230092.

Sort[Transpose[Select[Tally[Table[k + Total[IntegerDigits[k, 2]], {k, 0, 300}]], #[[2]] > 1 &]][[1]]] (* T. D. Noe, Oct 09 2013 *)

A228091 Numbers n for which there exists such a natural number k < n that k + bitcount(k) = n + bitcount(n), where bitcount(k) (A000120) gives the number of 1's in binary representation of nonnegative integer k.

Original entry on oeis.org

4, 12, 16, 17, 20, 28, 32, 34, 36, 44, 48, 49, 52, 60, 65, 68, 76, 80, 81, 84, 92, 96, 98, 100, 108, 112, 113, 116, 124, 128, 129, 130, 131, 132, 140, 144, 145, 148, 156, 160, 162, 164, 172, 176, 177, 180, 188, 193, 196, 204, 208, 209, 212, 220, 224, 226, 228

Offset: 1

Antti Karttunen, Aug 09 2013

nonn

In other words, all such terms A228236(n) which satisfy A228236(n) > A228086(A092391(A228236(n))), which means that the sequence contains all natural numbers n such that A228085(A092391(n)) > 1 and n > A228086(A092391(n)).

Note: 124 is the first term that occurs both here and in A228237.

			For cases 0 + A000120(0) = 0, 1 + A000120(1) = 2, 2 + A000120(2) = 3, 3 + A000120(3) = 5 there are no smaller solutions yielding the same result.
However, for 4 + A000120(4) = 5, we already saw the case 3+A000120(3) giving the same result, thus 4 is the first term of this sequence.
Next time this occurs for 12, as 12 + A000120(12) = 14 = 11 + A000120(11), and 11 < 12.

Antti Karttunen, Table of n, a(n) for n = 1..10000
Index entries for Colombian or self numbers and related sequences

Subset of A228236. Cf. also A228237. Complement of this sequence gives the nonzero terms of A228086 in ascending order.

Cf. A228085, A228086, A092391.

A242403 Decimal expansion of the binary self-numbers density constant.

Original entry on oeis.org

2, 5, 2, 6, 6, 0, 2, 5, 9, 0, 0, 8, 8, 8, 2, 9, 2, 2, 1, 5, 5, 0, 6, 2, 7, 1, 4, 3, 2, 7, 8, 9, 4, 1, 4, 1, 8, 2, 5, 2, 1, 9, 3, 3, 9, 6, 2, 9, 7, 8, 4, 6, 1, 3, 0, 1, 6, 8, 6, 2, 1, 7, 2, 2, 9, 2, 2, 8, 0, 5, 4, 8, 4, 4, 7, 6, 6, 3, 2, 5, 6, 6, 9, 5, 9, 1, 4, 2, 4, 4, 7, 9, 3, 8, 6, 8, 8, 9, 4, 9

Offset: 0

Jean-François Alcover, May 13 2014

This constant is transcendental (Troi and Zannier, 1999). - Amiram Eldar, Nov 28 2020

			0.2526602590088829221550627143278941418252...

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, p. 179.
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, Bollettino dell'Unione Matematica Italiana, Serie 7, Vol. 9-A, No. 1 (1995), pp. 143-148.

G. Troi and U. Zannier, Note on the density constant in the distribution of self-numbers - II, Bollettino dell'Unione Matematica Italiana, Serie 8, Vol. 2-B, No. 2 (1999), pp. 397-399; alternative link.
Umberto Zannier, On the distribution of self-numbers, Proc. Amer. Math. Soc., Vol. 85, No. 1 (1982), pp. 10-14.
Index entries for transcendental numbers

Cf. A010061 (binary self numbers), A003052 (decimal self numbers), A010064, A010067, A010070, A092391, A228082.

m0 = 100; dm = 100; digits = 100; Clear[lambda]; lambda[m_] := lambda[m] = Total[1/2^Union[Table[n + Total[IntegerDigits[n, 2]], {n, 0, m}]]]^2/8 // N[#, 2*digits]& // RealDigits[#, 10, 2*digits]& // First; lambda[m0]; lambda[m = m0 + dm]; While[lambda[m] != lambda[m - dm], Print["m = ", m]; m = m + dm]; lambda[m][[1 ;; digits]]

Equals (1/8)*(Sum_{n not a binary self-number} 1/2^n)^2.

cp's OEIS Frontend

A227915 Numbers of the form k + wt(k) for exactly four distinct k, where wt(k) = A000120(k) is the binary weight of k.

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A228086 a(n) is the least k which satisfies n = k + bitcount(k), or 0 if no such k exists. Here bitcount(k) (or wt(k), A000120) gives the number of 1's in binary representation of nonnegative integer k.

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A230091 Numbers of the form k + wt(k) for exactly two distinct k, where wt(k) = A000120(k) is the binary weight of k.

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A233272 a(n) = n + 1 + number of nonleading zeros in binary representation of n (A080791).

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A228087 a(n) = largest k which satisfies n = k + bitcount(k), or 0 if no such k exists. Here bitcount(k) (A000120) gives the number of 1's in binary representation of nonnegative integer k.

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A243441 Primes p such that p + A000120(p) is also a prime, where A000120 = sum of digits in base 2 = Hamming weight.

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A228083 Table of binary Self-numbers and their descendants; square array T(r,c), with row r>=1, column c>=1, read by antidiagonals.

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A230058 Numbers of the form k + wt(k) for at least two distinct k, where wt(k) = A000120(k) is the binary weight of k.

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