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 31-36 of 36 results.

A102682 Number of digits >= 8 in the decimal representations of all integers from 0 to n.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 4, 4, 4, 4, 4, 4, 4, 4, 4, 5, 6, 6, 6, 6, 6, 6, 6, 6, 6, 7, 8, 8, 8, 8, 8, 8, 8, 8, 8, 9, 10, 10, 10, 10, 10, 10, 10, 10, 10, 11, 12, 12, 12, 12, 12, 12, 12, 12, 12, 13, 14, 14, 14, 14, 14, 14, 14, 14, 14, 15, 16, 17, 18, 19, 20, 21, 22
Offset: 0

Views

Author

N. J. A. Sloane, Feb 03 2005

Keywords

Comments

The total number of digits >= 8 occurring in all the numbers 0, 1, 2, ... n (in decimal representation). - Hieronymus Fischer, Jun 10 2012

Links

Crossrefs

Partial sums of A102681.
Cf. A027868, A054899, A055640, A055641, A102669-A102685, A117804, A122840, A122841, A160093, A160094, A196563, A196564.
Cf. A000120, A000788, A023416, A059015 (for base 2).

Programs

  • Maple
    p:=proc(n) local b,ct,j: b:=convert(n,base,10): ct:=0: for j from 1 to nops(b) do if b[j]>=8 then ct:=ct+1 else ct:=ct fi od: ct: end: seq(add(p(i),i=0..n), n=0..95); # Emeric Deutsch, Feb 23 2005

Formula

From Hieronymus Fischer, Jun 10 2012: (Start)
a(n) = (1/2)*Sum_{j=1..m+1} (floor(n/10^j + 1/5)*(2n + 2 - (3/5 + floor(n/10^j + 1/5))*10^j) - floor(n/10^j)*(2n + 2 - (1+floor(n/10^j)) * 10^j)), where m = floor(log_10(n)).
a(n) = (n+1)*A102681(n) + (1/2)*Sum_{j=1..m+1} ((-3/5*floor(n/10^j + 1/5) + floor(n/10^j))*10^j - (floor(n/10^j + 1/5)^2 - floor(n/10^j)^2)*10^j), where m = floor(log_10(n)).
a(10^m-1) = 2*m*10^(m-1). (this is total number of digits >= 8 occurring in all the numbers with <= m places).
G.f.: g(x) = (1/(1-x)^2)*Sum_{j>=0} (x^(8*10^j) - x^(10*10^j))/(1-x^10^(j+1)). (End)

Extensions

More terms from Emeric Deutsch, Feb 23 2005
An incorrect g.f. was deleted by N. J. A. Sloane, Sep 16 2009

A220376 a(n) = position of first occurrence of n-th "early bird" number (cf. A116700) in the string 1234567891011121314151617181920212223242526272... .

Original entry on oeis.org

1, 15, 2, 17, 37, 3, 19, 39, 59, 4, 21, 41, 61, 81, 5, 23, 43, 63, 83, 103, 6, 25, 45, 65, 85, 105, 125, 7, 27, 47, 67, 87, 107, 127, 147, 8, 9, 29, 49, 69, 89, 109, 129, 149, 169, 10, 195, 12, 13, 14, 33, 1, 16, 53, 18, 73, 20, 93, 22, 113, 24, 133, 26, 153
Offset: 1

Views

Author

Reinhard Zumkeller, Dec 13 2012

Keywords

Examples

			.  1   5    10   15   20   25   30   35   40   45   50   55   60
.  ....v....x....v....x....v....x....v....x....v....x....v....x....
.  12345678910111213141516171819202122232425262728293031323334353...
.  |||           | | |                 | |
.  |||           | | |                 | |
.  |||           | | |                 | +---  A116700(8) = 42
.  |||           | | |                 |          -> a(8) = 39
.  |||           | | |                 |
.  |||           | | |                 +--- A116700(5) = 32
.  |||           | | |                         -> a(5) = 37
.  |||           | | |
.  |||           | | + --- A116700(7) = 41  -> a(7) = 19
.  |||           | |
.  |||           | +--- A116700(4) = 31  -> a(4) = 17
.  |||           |
.  |||           +--- A116700(2) = 21  -> a(2) = 15
.  |||
.  ||+--- A116700(6) = 34  -> a(6) = 3
.  ||
.  |+--- A116700(3) = 23  -> a(3) = 2
.  |
.  +--- A116700(1) = 12  -> a(1) = 1, see also A007908.

Links

Crossrefs

Cf. A117804.

Programs

  • Haskell
    import Data.List (isPrefixOf, find)
import Data.Maybe (fromJust)
a220376 n = a220376_list !! (n-1)
a220376_list = at 1 where
   at z | (reverse (show (z - 1)) `isPrefixOf` fst bird) = at (z + 1)
        | otherwise                = (length $ fst bird) : at (z + 1)
        where bird = fromJust $ find ((show z `isPrefixOf`) . snd) xys
   xys = iterate (\(us, v : vs) -> (v : us, vs))
                 ([], concatMap show [0 ..])

A100470 n appears A055642(n) times (appearances equal number of decimal digits).

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 10, 11, 11, 12, 12, 13, 13, 14, 14, 15, 15, 16, 16, 17, 17, 18, 18, 19, 19, 20, 20, 21, 21, 22, 22, 23, 23, 24, 24, 25, 25, 26, 26, 27, 27, 28, 28, 29, 29, 30, 30, 31, 31, 32, 32, 33, 33, 34, 34, 35, 35, 36, 36, 37, 37, 38, 38, 39, 39, 40, 40, 41
Offset: 0

Views

Author

Rick L. Shepherd, Nov 21 2004

Keywords

Comments

Lexicographically smallest nondecreasing left inverse to A117804: a(A117804(n)) = n. Gives the number to which belongs the digit A007376(n). - M. F. Hasler, Oct 23 2019

Crossrefs

Cf. A055642, A030530 (n's appearances equal its number of bits).

A132134 Base 3 "Punctual Bird" numbers: write the natural numbers, base 3, in a string 12101112202122100101102... Sequence gives numbers which do not occur in the string ahead of their natural place.

Original entry on oeis.org

1, 2, 3, 4, 6, 9, 11, 15, 18, 27, 29, 30, 32, 33, 35, 42, 45, 54, 60, 81, 83, 86, 87, 89, 92, 95, 96, 98, 99, 101, 104, 105, 107, 123, 126, 135, 141, 153, 162, 243, 245, 248, 249, 251, 252, 254, 257, 258, 260, 261, 263, 266, 267, 269, 275, 276, 278, 285, 287, 288
Offset: 1

Views

Author

Graeme McRae, Aug 11 2007

Keywords

Examples

			a(5)=6 because 6 (20, base 3) is the fifth number that appears first in its "natural" place in the string of concatenated base-3 numbers.

Crossrefs

Cf. A033307, A117804, A116700, A131881, A132131, A132132, A132133, A132135.

A383429 a(1)=1, and for n>1, a(n) = a(n-1) concatenated with the length of the decimal representation of a(n-1).

Original entry on oeis.org

1, 11, 112, 1123, 11234, 112345, 1123456, 11234567, 112345678, 1123456789, 112345678910, 11234567891012, 1123456789101214, 112345678910121416, 11234567891012141618, 1123456789101214161820, 112345678910121416182022, 11234567891012141618202224
Offset: 1

Views

Author

Jason Bard, Apr 27 2025

Keywords

Examples

			a(5) = 11234 and A055642(a(5)) = 5, so a(6) = 112345.

Links

Crossrefs

Cf. A007908, A055642, A117804.

Programs

  • Mathematica
    A[n_]:=NestList[#<>ToString[StringLength[#]]&,"1",n-1]; A[355]
  • PARI
    lista(nn) = my(va=vector(nn)); va[1] = 1; for (n=2, nn, va[n] = fromdigits(concat(digits(va[n-1]), digits(#Str(va[n-1]))));); va; \\ Michel Marcus, Apr 30 2025

A280724 Expansion of 1/(1 - x) + (1/(1 - x)^2)*Sum_{k>=0} x^(3^k).

Original entry on oeis.org

1, 2, 3, 5, 7, 9, 11, 13, 15, 18, 21, 24, 27, 30, 33, 36, 39, 42, 45, 48, 51, 54, 57, 60, 63, 66, 69, 73, 77, 81, 85, 89, 93, 97, 101, 105, 109, 113, 117, 121, 125, 129, 133, 137, 141, 145, 149, 153, 157, 161, 165, 169, 173, 177, 181, 185, 189, 193, 197, 201, 205, 209, 213, 217, 221, 225, 229, 233, 237, 241, 245
Offset: 0

Views

Author

Ilya Gutkovskiy, Jan 07 2017

Keywords

Comments

Sums of lengths of ternary numbers (A007089).

Examples

			-----------------------
n  base 3 length  a(n)
-----------------------
0 |  0   |  1   |  1
1 |  1   |  1   |  2
2 |  2   |  1   |  3
3 |  10  |  2   |  5
4 |  11  |  2   |  7
5 |  12  |  2   |  9
6 |  20  |  2   |  11
7 |  21  |  2   |  13
8 |  22  |  2   |  15
9 |  100 |  3   |  18
-----------------------

Links

  • Eric Weisstein's World of Mathematics, Ternary

Crossrefs

Cf. A007089, A062153, A081604, A083652, A117804.

Programs

  • Mathematica
    CoefficientList[Series[1/(1 - x) + (1/(1 - x)^2) Sum[x^3^k, {k, 0, 15}], {x, 0, 70}], x]
Table[1 + Sum[Floor[Log[3, k]] + 1, {k, 1, n}], {n, 0, 70}]

Formula

G.f.: 1/(1 - x) + (1/(1 - x)^2)*Sum_{k>=0} x^(3^k).
a(n) = 1 + Sum_{k=1..n} floor(log_3(k)) + 1.
Previous Showing 31-36 of 36 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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