A007088 -id:A007088 - cp's OEIS frontend

A304747 May code shown in binary: a(n) = A007088(A303767(n)).

0, 1, 11, 10, 110, 100, 101, 111, 1111, 1000, 1001, 1011, 1010, 1110, 1100, 1101, 11101, 10000, 10001, 10011, 10010, 10110, 10100, 10101, 10111, 11111, 11000, 11001, 11011, 11010, 11110, 11100, 111100, 100000, 100001, 100011, 100010, 100110, 100100, 100101, 100111, 101111, 101000, 101001, 101011, 101010, 101110, 101100, 101101

Offset: 0

Antti Karttunen, May 23 2018

			The code can be constructed by the rule: a(n+1) is either the least number obtained from a(n) by toggling one or more 1-bits off if no such number is yet in the sequence, otherwise the least number not yet in sequence that can be obtained from a(n) by toggling one 0-bit on:
   n    a(n)
   0      0
   1      1
   2     11
   3     10
   4    110
   5    100
   6    101
   7    111
   8   1111
   9   1000
  10   1001
  11   1011
  12   1010
  13   1110
  14   1100
  15   1101
  16  11101
  17  10000
  18  10001
  19  10011
  20  10010
  21  10110
  22  10100
  23  10101
  24  10111
  25  11111
  26  11000
  27  11001
  28  11011
  29  11010
  30  11110
  31  11100
  32 111100
  33 100000

Antti Karttunen, Table of n, a(n) for n = 0..8191
Index entries for sequences related to binary expansion of n

Cf. A007088, A303767.

Cf. also A304749.

A209229(n) = (n && !bitand(n,n-1));
A053644(n) = { my(k=1); while(k<=n, k<<=1); (k>>1); }; \\ From A053644
A303767(n) = if(!n,n,if(A209229(n),n+A303767(n-1),A053644(n)+A303767(n-A053644(n)-1)));
A007088(n) = fromdigits(binary(n), 10); \\ From A007088.
A304747(n) = A007088(A303767(n));

a(n) = A007088(A303767(n)).

A339544 Primes p such that A007088(p) mod p is prime.

Original entry on oeis.org

3, 17, 19, 29, 31, 71, 79, 83, 103, 113, 151, 211, 229, 293, 331, 337, 347, 349, 421, 439, 449, 457, 607, 659, 683, 691, 739, 743, 809, 839, 883, 911, 977, 1039, 1051, 1193, 1249, 1277, 1283, 1303, 1367, 1439, 1451, 1499, 1567, 1597, 1609, 1663, 1747, 1753, 1861, 2089, 2137, 2273, 2309, 2311

Offset: 1

J. M. Bergot and Robert Israel, Dec 08 2020

			a(3) = 19 because 19 = 10011_2 and 10011 mod 19 = 17 which is prime.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A007088.

filter:= proc(n) isprime(n) and isprime(convert(n,binary) mod n) end proc:
select(filter, [seq(i,i=3..10000,2)]);

A084484 a(n) = A007088(A084483(n)).

Original entry on oeis.org

1, 100, 11, 10, 1001, 1100, 111, 10000, 101, 10100, 10011, 110, 11001, 11100, 1111, 1000, 100001, 100100, 1011, 1010, 101001, 101100, 100111, 110000, 1101, 110100, 110011, 1110, 111001, 111100, 11111, 1000000, 10001, 1000100, 1000011, 10010, 1001001, 1001100

Offset: 1

Reinhard Zumkeller, May 27 2003

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000120, A007088, A023416, A070939, A084483.

s[n_] := s[n] = If[OddQ[n], 2*s[(n - 1)/2] + 1, If[EvenQ[IntegerExponent[n, 2]], n/2, 2*n]]; FromDigits[IntegerDigits[#, 2]]& /@ Array[s, 50] (* Amiram Eldar, Jul 22 2023 *)

a(n) = A007088(n) iff A000120(n) = A070939(n).

A138342 First differences of A007088.

Original entry on oeis.org

1, 9, 1, 89, 1, 9, 1, 889, 1, 9, 1, 89, 1, 9, 1, 8889, 1, 9, 1, 89, 1, 9, 1, 889, 1, 9, 1, 89, 1, 9, 1, 88889, 1, 9, 1, 89, 1, 9, 1, 889, 1, 9, 1, 89, 1, 9, 1, 8889, 1, 9, 1, 89, 1, 9, 1, 889, 1, 9, 1, 89, 1, 9, 1, 888889, 1, 9, 1, 89, 1, 9, 1, 889, 1, 9, 1, 89, 1, 9, 1, 8889, 1, 9, 1, 89, 1, 9

Offset: 1

Jaume Simon Gispert (jaume(AT)nuem.com), May 17 2008

			1-0 = 1, 10-1 = 9, 11-10 = 1, 100-11 = 89, ...

Antti Karttunen, Table of n, a(n) for n = 1..16383

Cf. A007088, A007814, A059482.

Differences[Table[FromDigits[IntegerDigits[n,2]],{n,0,90}]] (* Harvey P. Dale, Feb 26 2012 *)

A007088(n) = fromdigits(binary(n), 10); \\ From A007088.
A138342(n) = (A007088(n) - A007088(n-1)); \\ Antti Karttunen, Nov 06 2018

A059482(n) = ((10^n)*(1000/1125) + (1/9));
A138342(n) = { my(f=factor(n)); prod(i=1,#f~,if(2==f[i,1],A059482(f[i,2]),1)); }; \\ Antti Karttunen, Nov 06 2018

a(n) = A059482(A007814(n)).
From Antti Karttunen, Nov 06 2018: (Start)
a(n) = A007088(n) - A007088(n-1).
Multiplicative with a(2^e) = A059482(e), a(p^e) = 1 for odd primes p.
(End)
G.f.: Sum_{k>=0} 10^k * x^(2^k) / (1 + x^(2^k)). - Ilya Gutkovskiy, Dec 14 2020

Offset corrected and keyword:mult added by Antti Karttunen, Nov 06 2018

A154474 a(n) = A007088(A154473(n)).

Original entry on oeis.org

1101001010, 10101101010010, 11001110010010, 1011011100100010, 1110111001000010, 1101110010001110111001000100, 10111001001101110010011101110010001000

Offset: 0

Antti Karttunen, Jan 11 2009

This sequence gives the parenthesis expressions shown at the upper right corner image of the page 103 of NKS, with left brackets (black squares) converted to 1's and right brackets (white squares) converted to 0's. Compare to A080070, A122242, A122245.

S. Wolfram, A New Kind of Science, Wolfram Media Inc., (2002), p. 102, p. 103 and pages 104, 896-898.

A339567 Numbers k such that A007088(k) == 1 (mod k).

Original entry on oeis.org

1, 5, 15, 25, 55, 91, 137, 525, 625, 925, 3967, 5995, 7625, 10767, 25087, 57225, 68817, 565027, 591415, 2515825, 2757625, 4162019, 5276309, 96689255, 115686005, 133890625, 242899421, 492029715, 588620625, 1839399055, 7786281065, 11231388063, 17251448809, 71050380625

Offset: 1

Robert Israel, Dec 09 2020

All terms are odd.

			a(3) = 15 is a term because 15 = 1111_2 and 1111 == 1 (mod 15).

Cf. A007088, A339566.

filter:= t -> convert(t,binary) mod t = 1: filter(1):= true:
select(filter, [seq(i,i=1..10^7,2)]);

Block[{a = {1}, k}, Do[If[Mod[FromDigits@ IntegerDigits[i, 2], i] == 1, AppendTo[a, i]], {i, 2, 10^7}]; a] (* Michael De Vlieger, Dec 12 2020 *)

isok(n) = Mod(fromdigits(binary(n)), n) == 1;
forstep(k=1, 10^7, 2, if(isok(k), print1(k, ", "))); \\ Daniel Suteu, Dec 12 2020

a(30)-a(34) from Daniel Suteu, Dec 12 2020

A354837 Odd numbers k such that gcd(k, A007088(k)) != 1.

Original entry on oeis.org

21, 33, 63, 69, 81, 99, 111, 123, 159, 165, 183, 189, 203, 207, 219, 231, 237, 243, 249, 259, 261, 273, 297, 303, 315, 321, 363, 399, 411, 423, 429, 435, 441, 459, 483, 489, 495, 513, 543, 561, 567, 573, 585, 591, 603, 615, 621, 627, 633, 663, 669, 693, 707, 711

Offset: 1

Ctibor O. Zizka, Jun 08 2022

			k = 33; gcd(33,A007088(33)) = gcd(33, 100001) = 11, thus k = 33 is a term.

Cf. A007088, A354838.

Select[Range[1, 700, 2], ! CoprimeQ[#, FromDigits[IntegerDigits[#, 2]]] &] (* Amiram Eldar, Jun 08 2022 *)

isok(k) = (k%2) && (gcd(k, fromdigits(binary(k))) != 1); \\ Michel Marcus, Jun 09 2022

A020767 Product_{k=1..n} b(k), where b(k) = binary expansion of k (A007088) but read as if it were a decimal number.

Original entry on oeis.org

1, 1, 10, 110, 11000, 1111000, 122210000, 13565310000, 13565310000000, 13578875310000000, 13714664063100000000, 13865525367794100000000, 15252077904573510000000000, 16792537772935434510000000000, 18639716927958332306100000000000, 20708725506961707192077100000000000

Offset: 0

Susanna Cuyler, May 23 2003

			a(4) = 1*10*11*100 = 11000.

Partial products of A007088.

A140117 Numbers n for which A140116(n) uses fewer symbols than A007088(n).

Original entry on oeis.org

2, 4, 8, 15, 16, 31, 32, 33, 34, 63, 64, 65, 66, 68, 72, 125, 126, 127, 128, 129, 130, 131, 132, 136, 144, 160, 192, 247, 251, 253, 254, 255, 256, 257, 258, 259, 260, 264, 272, 288, 320, 384, 503, 507, 509, 510, 511, 512, 513, 514, 515, 516, 517, 518, 520, 521

Offset: 1

Rick L. Shepherd, May 08 2008

nonn

The representation of n given by A140116(n) is more compact than the standard binary form of n for these numbers consisting of a sufficiently large proportion of 0's or of 1's among their bits.

			See also examples for A140116. 2180 is a term here as 2180 = 100010000100 (base 2), which uses 12 symbols, but A140116(2180) = 10112111210, which uses only 11 symbols including the two delimiter symbols (for which 2 is used).

Cf. A140116, A007088.

A186951 Number of lunar divisors (in base 10) of the n-th nonzero number whose decimal expansion contains only 0's and 1's (A007088(n)).

Original entry on oeis.org

9, 18, 90, 27, 90, 180, 819, 36, 90, 180, 738, 270, 738, 1638, 7461, 45, 90, 180, 738, 270, 819, 1476, 6570, 360, 738, 1476, 6732, 2457, 6570, 14922, 67968, 54, 90, 180, 738, 270, 738, 1476, 6570, 360, 738, 1638, 6570, 2214, 6732, 13140, 59868, 450, 738, 1476, 6732, 2214, 6570, 13464, 59868, 3276, 6570, 13140, 59868, 22383, 59868, 135936, 619902

Offset: 1

N. J. A. Sloane, Mar 01 2011

N. J. A. Sloane, Table of n, a(n) for n = 1..99
D. Applegate, M. LeBrun and N. J. A. Sloane, Dismal Arithmetic [Note: we have now changed the name from "dismal arithmetic" to "lunar arithmetic" - the old name was too depressing]
Index entries for sequences related to dismal (or lunar) arithmetic

A007088, A087029, A186943.

cp's OEIS Frontend

A304747 May code shown in binary: a(n) = A007088(A303767(n)).

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A339544 Primes p such that A007088(p) mod p is prime.

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A084484 a(n) = A007088(A084483(n)).

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A138342 First differences of A007088.

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A154474 a(n) = A007088(A154473(n)).

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A339567 Numbers k such that A007088(k) == 1 (mod k).

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A354837 Odd numbers k such that gcd(k, A007088(k)) != 1.

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A020767 Product_{k=1..n} b(k), where b(k) = binary expansion of k (A007088) but read as if it were a decimal number.

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A140117 Numbers n for which A140116(n) uses fewer symbols than A007088(n).

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