cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Previous Showing 31-34 of 34 results.

A378332 Decimal expansion of the base 8 Champernowne constant.

Original entry on oeis.org

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

Views

Author

Joshua Searle, Nov 25 2024

Keywords

Comments

This constant is formed by the concatenation of the natural numbers in base 8 and then converted into base 10.
This constant is 8-normal.

Examples

			0.163264812105216797367094986142605190224237843285462333081380700428319475...

Links

Crossrefs

Cf. A030302, A003137, A030373, A031219, A030548, A030998, A054634, A031076, A033307 (base n expansions of base n Champernowne constants, without leading zero, for 2 <= n <= 10).
Cf. A066716, A077771, A378328, A378329, A378330, A378331, A378332, A378333, A033307 (decimal expansions of base n Champernowne constants for 2 <= n <= 10).
Cf. A066717, A077772, A378345, A378346, A378347, A378348, A378349, A378350, A030167 (continued fraction expansions of base n Champernowne constants for 2 <= n <= 10).

Programs

  • Mathematica
    First[RealDigits[ChampernowneNumber[8], 10, 100]]

A378333 Decimal expansion of the base 9 Champernowne constant.

Original entry on oeis.org

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

Views

Author

Joshua Searle, Nov 25 2024

Keywords

Comments

This constant is formed by the concatenation of the natural numbers in base 9 and then converted into base 10.
This constant is 9-normal.

Examples

			0.140624976119696782479669008935663183265457083246828486657555171275414914...

Links

Crossrefs

Cf. A030302, A003137, A030373, A031219, A030548, A030998, A054634, A031076, A033307 (base n expansions of base n Champernowne constants, without leading zero, for 2 <= n <= 10).
Cf. A066716, A077771, A378328, A378329, A378330, A378331, A378332, A378333, A033307 (decimal expansions of base n Champernowne constants for 2 <= n <= 10).
Cf. A066717, A077772, A378345, A378346, A378347, A378348, A378349, A378350, A030167 (continued fraction expansions of base n Champernowne constants for 2 <= n <= 10).

Programs

  • Mathematica
    First[RealDigits[ChampernowneNumber[9], 10, 100]]

A355487 Bitwise XOR of the base-4 digits of n.

Original entry on oeis.org

0, 1, 2, 3, 1, 0, 3, 2, 2, 3, 0, 1, 3, 2, 1, 0, 1, 0, 3, 2, 0, 1, 2, 3, 3, 2, 1, 0, 2, 3, 0, 1, 2, 3, 0, 1, 3, 2, 1, 0, 0, 1, 2, 3, 1, 0, 3, 2, 3, 2, 1, 0, 2, 3, 0, 1, 1, 0, 3, 2, 0, 1, 2, 3, 1, 0, 3, 2, 0, 1, 2, 3, 3, 2, 1, 0, 2, 3, 0, 1, 0, 1, 2, 3, 1, 0, 3
Offset: 0

Views

Author

Kevin Ryde, Jul 04 2022

Keywords

Comments

Equivalently, the parity of the odd position 1-bits of n and the parity of the even position 1-bits of n, combined as a(n) = 2*A269723(n) + A341389(n).
In GF(2)[x] polynomials encoded as bits of an integer (least significant bit for the constant term), a(n) is remainder n mod x^2 + 1.

Examples

			n=35 has base-4 digits 203 so a(35) = 2 XOR 0 XOR 3 = 1.

Links

Crossrefs

Cf. A030373 (base 4 digits), A003987 (XOR).
Cf. A341389, A269723, A010060.
Cf. A353167 (indices of 0's).
Other digit operations: A053737 (sum), A309954 (product).

Programs

  • Mathematica
    a[n_] := BitXor @@ IntegerDigits[n, 4]; Array[a, 100, 0] (* Amiram Eldar, Jul 05 2022 *)
  • PARI
    a(n) = if(n==0,0, fold(bitxor,digits(n,4)));
  • Python
    from operator import xor
from functools import reduce
from sympy.ntheory import digits
def a(n): return reduce(xor, digits(n, 4)[1:])
print([a(n) for n in range(87)]) # Michael S. Branicky, Jul 05 2022

Formula

Fixed point of the morphism 0 -> 0,1; 1 -> 2,3; 2 -> 1,0; 3 -> 3,2 starting from 0.

A275993 Champernowne sequence: write n in base 16 and juxtapose.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 1, 0, 1, 1, 1, 2, 1, 3, 1, 4, 1, 5, 1, 6, 1, 7, 1, 8, 1, 9, 1, 10, 1, 11, 1, 12, 1, 13, 1, 14, 1, 15, 2, 0, 2, 1, 2, 2, 2, 3, 2, 4, 2, 5, 2, 6, 2, 7, 2, 8, 2, 9, 2, 10, 2, 11, 2, 12, 2, 13, 2, 14, 2, 15, 3, 0, 3, 1, 3, 2, 3, 3, 3, 4, 3, 5, 3, 6, 3, 7, 3, 8, 3, 9, 3, 10, 3, 11, 3
Offset: 0

Views

Author

Robert G. Wilson v, Aug 15 2016

Keywords

Comments

10 -> A, 11 -> B, 12 -> C, 13 -> D, 14 -> E & 15 -> F.

Links

Crossrefs

Tables in which the n-th row lists the base b digits of n: A030190 and A030302 (b=2), A003137 and A054635 (b=3), A030373 (b=4), A031219 (b=5), A030548 (b=6), A030998 (b=7), A031035 and A054634 (b=8), A031076 (b=9), A007376 and A033307 (b=10).

Programs

  • Mathematica
    almostNatural[n_, b_] := Block[{m = 0, d = n, i = 1, l, p}, While[m <= d, l = m; m = (b -1) i*b^(i -1) + l; i++]; i--; p = Mod[d -l, i]; q = Floor[(d -l)/i] + b^(i -1); If[p != 0, IntegerDigits[q, b][[p]], Mod[q -1, b]]]; Array[ almostNatural[#, 16] &, 105, 0]
First[RealDigits[ChampernowneNumber[16], 16, 100, 0]] (* Paolo Xausa, Jun 21 2024 *)
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