Displaying 1-10 of 15 results found.
Let d = A096567(n) be the first digit to appear n times in the decimal expansion of Pi; if d is the m-th digit of Pi, a(n) = m.
+20
2
1, 4, 11, 18, 25, 26, 28, 44, 47, 59, 63, 80, 81, 101, 108, 114, 125, 135, 148, 151, 153, 162, 172, 187, 198, 205, 206, 223, 229, 234, 237, 256, 268, 274, 279, 294, 297, 304, 322, 335, 338, 355, 374, 381, 387, 393, 401, 433, 438, 439, 443, 446, 447, 472, 484, 491, 495
EXAMPLE
Pi = 3.14159265358979323...
The first digit to appear 4 times in Pi is 3, at the 18th digit, so a(4) = 18.
PROG
(PARI) See Links section.
First digit to appear n times in the decimal expansion of e.
+10
10
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
COMMENTS
The digits 0 and 5 do not appear among the first 30000 terms. When do they first appear? - Jianing Song, Apr 01 2021
EXAMPLE
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)
MATHEMATICA
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 *)
First 2-digit number to appear n times in the decimal expansion of Pi.
+10
9
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
COMMENTS
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 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)
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
REFERENCES
William Feller, An Introduction to Probability Theory and Its Applications, Vol. I, Chapter III, Wiley, 3rd Ed., Corrected printing 1970.
EXAMPLE
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.
MATHEMATICA
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 *)
Leaders in the race of digits of Pi.
+10
8
3, 1, 5, 3, 9, 8, 2, 8, 4, 8, 2, 8, 2, 4, 1, 9, 1, 9, 1, 9, 1, 9, 1, 5, 1, 5, 1, 5, 1, 5, 1, 5, 1, 5, 1, 5, 1, 5, 1, 5, 1, 5, 1, 5, 1, 5, 1, 5, 1, 5, 1, 5, 1, 5, 1, 5, 4, 5, 4, 5, 4, 5, 4, 5, 4, 5, 4, 5, 4, 5, 4, 5, 4, 5, 4, 5, 4, 5, 4, 5, 4, 5, 4, 5, 4, 1, 4
COMMENTS
Next term which is different from earlier in A096567. The number 4 wins 71.7% of the first 100 million races (occurs most often in 71.7% of the races). It is also the leader after 100 million digits with a comfortable lead (10,003,863 occurrences compared to 10,002,475 occurrences of the 1 that was winning 15.9% of the first 100 million races). All numbers except the 6 were in the lead at some time. Number 6 was almost in the lead after 48,500 digits, only two occurrences short of the 1 at that time. In the first 100,000,000 digits of Pi the number 6 appears about 4450 times less than the current leader 4. But as the next comment shows the 6 finally takes the lead after 990,213,634 digits. - Ruediger Jehn, Jan 27 2021 Position at which a number (0 to 9) is leader for the first time: 174999, 4, 187, 1, 274, 11, 990213634, 320741, 108, 59 (see A342325). - Kester Habermann, Jan 27 2021
EXAMPLE
The decimal expansion of Pi = 3.1415926535... starts with 3 (see A000796) hence the first leader in the race of digits is 3, so a(1) = 3. After 4 stages the new leader is 1 because the number 1 appears twice and the earlier leader appears once, so a(2) = 1. After 11 stages the new leader is 5 because the number 5 appears three times and the earlier leader appears twice, so a(3) = 5.
First 3-digit number to appear n times in the decimal expansion of Pi.
+10
8
314, 592, 446, 117, 105, 19, 381, 279, 609, 609, 848, 848, 654, 654, 654, 654, 19, 19, 965, 965, 965, 965, 19, 19, 19, 494, 564, 390, 390, 390, 390, 390, 682, 682, 390, 346, 390, 390, 390, 390, 390, 390, 346, 346, 346, 99, 201, 201, 201, 201, 201, 201, 201
COMMENTS
a(6) is the 3-digit number 019.
By the pigeonhole principle, it suffices to examine 1000n - 997 digits of Pi to find the n-th term; on average 1000n - O(sqrt n) will suffice. Do each of 0..999 appear in this sequence? Which appears last? - Charles R Greathouse IV, Sep 26 2016
EXAMPLE
a(2) = 592 because 592 is the first 3-digit number to appear 2 times in the decimal expansion of Pi = 3.141(592)653589793238462643383279502884197169399375105820974944(592)...
First digit to appear n times in the decimal expansion of the golden ratio phi.
+10
7
1, 1, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 2, 6, 2, 8, 8, 8, 8, 8, 8, 8, 8, 4, 4, 8, 8, 8, 8, 8, 8, 8, 8, 8, 4, 4, 6, 6, 4, 4, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 6, 6, 6, 2, 2
First 4-digit number to appear n times in the decimal expansion of Pi.
+10
7
3141, 582, 9999, 2796, 6549, 2019, 2916, 8352, 5485, 5485, 5485, 5485, 5485, 5485, 5485, 1177, 1177, 5485, 1177, 3718, 5485, 5485, 1766, 1766, 5485, 4608, 4608, 4608, 4608, 4608, 5485, 5485, 504, 504, 504, 504, 504, 504, 504, 2103, 504, 504, 9479, 504, 504
COMMENTS
Note that the sequence contains elements whose number of digits is less than 4. See example.
The first 3 appearances of a(3) = 9999 overlap in 999999.
EXAMPLE
a(2) = 582 because "0582" is the first 4-digit number to appear 2 times in the decimal expansion of Pi = 3.1415926535897932384626433832795028841971693993751(0582)097494459230781\ 640628620899862803482534211706798214808651328230664709384460955(0582)...
First 5-digit number to appear n times in the decimal expansion of Pi.
+10
6
31415, 60943, 48940, 36041, 86538, 85990, 40230, 91465, 26063, 87258, 87258, 87258, 56517, 15157, 47392, 15157, 87258, 87258, 15157, 15157, 46083, 46083, 46083, 46083, 15931, 15931, 10767, 10767, 10767, 18804, 18804, 83903, 83903, 83903, 18271, 83903, 83903
EXAMPLE
a(2) = 60943 because 60943 is the first 5-digit number to appear 2 times in the decimal expansion of Pi.
First 6-digit number to appear n times in the decimal expansion of Pi.
+10
6
314159, 949129, 266830, 178653, 872117, 872117, 872117, 919441, 919441, 735287, 820737, 420516, 802307, 556505, 267638, 107072
COMMENTS
The first 2 appearances of a(2) = 949129 both end at the beginning of a block of 100 digits of Pi after the decimal point. The 5th block of 100 digits of Pi after the decimal point ends with 94912, and the 6th block of 100 digits starts with 9. The 13th block of 100 digits of Pi after the decimal point ends with 94912, and the 14th block of 100 digits starts with 9.
EXAMPLE
a(2) = 949129 because 949129 is the first 6-digit number to appear 2 times in the decimal expansion of Pi.
First 10-digit number to appear n times in the decimal expansion of Pi.
+10
6
3141592653, 4392366484, 9526413073, 7454969632, 1459184231, 3955267283
COMMENTS
This sequence was mentioned in a forum post called "Ten repeating numbers in Pi". It was about finding a 10-digit number that repeated in Pi. The answer was a(2) = A197123(10) = 4392366484.
EXAMPLE
a(2) = 4392366484 because 4392366484 is the first 10-digit number to appear 2 times in the decimal expansion of Pi.
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