A277270 -id:A277270 - cp's OEIS frontend

A195138 First digit to appear n times in the decimal expansion of e.

2, 2, 8, 8, 2, 2, 2, 2, 9, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 2, 2, 2, 2, 2, 2, 2, 3, 9, 9, 2, 7, 4, 4, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9

Offset: 1

Omar E. Pol, Oct 22 2011

The digits 0 and 5 do not appear among the first 30000 terms. When do they first appear? - Jianing Song, Apr 01 2021

			From _Michael De Vlieger_, Sep 10 2017: (Start)
a(n) is the first decimal digit of e that first appears n times when e is expanded to the -m place:
   n  a(n)  m
   1   2    0
   2   2    4
   3   8    7
   4   8    9
   5   2   22
   6   2   30
   7   2   33
   8   2   40
   9   9   58
  10   7   63
  11   7   64
  12   7   68
  13   7   78
  14   7   83
  15   7   89
  16   7   99
(End)

Alois P. Heinz, Table of n, a(n) for n = 1..10000

Cf. A001113, A195844, A277270, A290643, A290644.

Cf. A096567, A195139, A195141, A280811, A341438.

With[{e = First@ RealDigits[N[E, 10^4]]}, Function[t, -1 + Map[FirstPosition[t, #] &, Range@ Max@ t][[All, -1]]]@ Table[BinCounts[Take[e, n], {0, 10, 1}], {n, 10^3}]] (* Michael De Vlieger, Sep 10 2017 *)

More terms from D. S. McNeil, Oct 22 2011

A276992 First 2-digit number to appear n times in the decimal expansion of Pi.

Original entry on oeis.org

31, 26, 93, 62, 82, 28, 28, 28, 48, 48, 48, 48, 48, 9, 9, 81, 17, 17, 95, 95, 95, 95, 95, 95, 95, 19, 21, 21, 21, 19, 95, 9, 9, 9, 95, 46, 95, 59, 9, 9, 9, 95, 95, 95, 95, 59, 59, 59, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 14, 14, 14, 9, 9, 9, 9, 14, 9, 9

Offset: 1

Bobby Jacobs, Sep 24 2016

a(n) is the 2-digit number that appears in Pi n times before any other 2-digit number appears in Pi n times.
Note that the sequence contains elements whose number of digits is 2 or 1, see examples. - Omar E. Pol, Oct 05 2016
Comment from N. J. A. Sloane, Mar 08 2023 (Start)
Make a table T[0,0], T[0,1], ...,T[9,9], with 100 columns, labeled 0,0 to 9,9.
Scan the digits of pi = 3.14159....
First you see 3, 1 so increment the count for 3,1; next you see 1,4, so increment the count for 1,4.  Then you see 4,1 so increment the count for 4,1.  Do this for ever.
The first time any count hits 6, say T[3,8] = 6, then a(6) = 38.
If it happens that T[0,9] hits 6 first, then a(6) would be 09, but we would drop the 0, and write a(6) = 9.
(End)
Comment from Alois P. Heinz, Mar 08 2023 (Start)
Initially, "09" is very often the first to occur n times, while other 2-digit  substrings fall behind. They can show up later.  This is not strange, this is Pi.
In the first 10000 terms we see "09" 40 times, "14" 33 times, and so on.  Here is the complete list:
[40, 9], [33, 14], [2, 17], [13, 19], [3, 21], [1, 26], [892, 27], [3, 28], [1, 31], [144, 34], [107, 35], [179, 39], [2594, 46], [5, 48], [127, 54], [1387, 55], [4, 59], [6, 62], [41, 65], [671, 71], [19, 74], [3406, 76], [1, 81], [1, 82], [94, 85], [1, 93], [211, 94], [14, 95].
67 of the two-digit strings never show up in the first 10000 terms.
It does not mean that they do not appear in Pi.  Indeed they do.  It only means that they are never the first to reach some count.  They may be behind by only a small amount. (End)
The fact that 09 is ahead so often is an example of the Arcsine Law Paradox at work. See for example Feller, Volume I, Chapter III.  As Feller says, "[the conclusions] are not only unexpected but actually come as a shock to intuition and common sense." Of course the same phenomenon occurs with single digits of Pi, see A096567, where 5 seems to be ahead most of the time. - N. J. A. Sloane, Mar 09 2023

			a(2) = 26 because 26 is the first 2-digit number to appear 2 times in the decimal expansion of Pi = 3.14159(26)5358979323846(26)...
a(14) = 9 because "09" is the first 2-digit number to appear 14 times in the decimal expansion of Pi.

William Feller, An Introduction to Probability Theory and Its Applications, Vol. I, Chapter III, Wiley, 3rd Ed., Corrected printing 1970.

Alois P. Heinz, Table of n, a(n) for n = 1..10000

Cf. A000796, A096567, A276686, A276993, A277171, A277270, A291599, A291600.

spi = ToString[Floor[10^100000 Pi]]; f[n_] := Block[{k = 2}, While[Length@ StringPosition[ StringTake[spi, k], StringTake[spi, {k - 1, k}]] != n, k++]; ToExpression@ StringTake[spi, {k - 1, k}]]; Apply[f, 72] (* Robert G. Wilson v, Oct 05 2016 *)

a(21)-a(40) from Bobby Jacobs, Oct 01 2016

More terms from Alois P. Heinz, Oct 02 2016

A290643 First 3-digit number to appear n times in the decimal expansion of e.

Original entry on oeis.org

271, 182, 75, 499, 793, 320, 320, 23, 23, 23, 23, 23, 23, 23, 23, 709, 709, 709, 709, 709, 171, 171, 171, 171, 171, 171, 171, 171, 171, 166, 166, 166, 93, 772, 772, 232, 232, 232, 232, 232, 772, 772, 772, 772, 772, 772, 772, 772, 772, 772, 772, 772, 772, 772, 772, 772, 772, 772, 772, 772

Offset: 1

Bobby Jacobs, Aug 08 2017

Some of the numbers start with 0. For example, a(3) is the 3-digit number 075.

			a(2) = 182 because 182 is the first 3-digit number to appear 2 times in the decimal expansion of e = 2.7(182)8(182)...

Cf. A001113, A195138, A276993, A277270, A290644.

A290644 First 4-digit number to appear n times in the decimal expansion of e.

Original entry on oeis.org

2718, 1828, 8793, 8793, 7093, 7093, 7093, 7093, 7093, 7093, 7093, 7093, 7093, 7093, 352, 352, 235, 235, 235, 352, 352, 352, 352, 352, 352, 1661, 352, 352, 352, 352, 352, 1891, 1891, 1891, 1891, 1891, 1891, 352, 3917, 3917, 3917, 3917, 3917, 3917, 3917, 5065, 8149, 8149, 8149, 8149

Offset: 1

Bobby Jacobs, Aug 08 2017

Some of the numbers start with 0. For example, a(15) is the 4-digit number 0352.

The first two appearances of a(2) = 1828 appear consecutively as 18281828.

			a(2) = 1828 because 1828 is the first 4-digit number to appear 2 times in the decimal expansion of e = 2.7(1828)(1828)...

Cf. A001113, A195138, A276686, A277270, A290643.

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A195138 First digit to appear n times in the decimal expansion of e.

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A276992 First 2-digit number to appear n times in the decimal expansion of Pi.

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A290643 First 3-digit number to appear n times in the decimal expansion of e.

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A290644 First 4-digit number to appear n times in the decimal expansion of e.

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