A078197 -id:A078197 - cp's OEIS frontend

A014777 Position of the start of the first occurrence of n after the decimal point in Pi = 3.14159265358979323846264338327950288...

32, 1, 6, 9, 2, 4, 7, 13, 11, 5, 49, 94, 148, 110, 1, 3, 40, 95, 424, 37, 53, 93, 135, 16, 292, 89, 6, 28, 33, 186, 64, 137, 15, 24, 86, 9, 285, 46, 17, 43, 70, 2, 92, 23, 59, 60, 19, 119, 87, 57, 31, 48, 172, 8, 191, 130, 210, 404, 10, 4, 127, 219, 20, 312, 22, 7, 117, 98, 605

Offset: 0

Paul Simon (paulsimn(AT)microtec.net) and Simon Plouffe

This is A037008(1), A037000(1), A037001(1), A037002(1), A037003(1), A037004(1), A037005(1), A036974(1), A037006(1), A037007(1) etc.

			In the decimal expansion of Pi, the string "0" is found at position 32 counting from the first digit after the decimal point. The string "1" is found at position 1, the string "2" at position 6, the string "3" at position 9, etc.

Ronald R. King and T. D. Noe, Table of n, a(n) for n = 0..9999
Dave Andersen, The Pi-Search Page.
Tom Crawford and Brady Haran, Strings and Loops within Pi, Numberphile video (2020)
Anders Hellström, Sage program, Feb 02 2017

Cf. A000796, A078197, A103186 (another version).

Cf. A037008, A037000, A037001, A037002, A037003, A037004, A037005, A036974, A037006, A037007.

k := 700; R := RealField(k); [ Position(IntegerToString(Round(10^k*(-3 + Pi(R)))), IntegerToString(n)) : n in [0..68] ]; /* Klaus Brockhaus, Feb 15 2007 */

Table[-1 + SequencePosition[#, IntegerDigits@ n][[1, 1]], {n, 0, 68}] &@ First@ RealDigits@ N[Pi, 10^4] (* Michael De Vlieger, Aug 10 2016, Version 10.1 *)

M14777=Map(); A014777(n)={iferr(mapget(M14777, n), E, my(i=if(n>9, A014777(n\10), 1), d=if(n, digits(n), [0]), j); while(i++, j=#d; until(!j, d[j]==A000796(i+j--) || next(2)); break); mapput(M14777, n, i--); i)} \\ M. F. Hasler, Jun 21 2022

from mpmath import mp
def A014777(n):
    if not (i := A014777.pos.get(n, 0)):
        d = str(n); s = 2 # starting position for search
        while (i := A014777.pi.find(d, s)) < 1:
            s = max(len(A014777.pi) - len(d), 2)
            with mp.workdps(s + 99 if s < 500 else s*6//5): # new precision
                A014777.pi = str(mp.pi - 5/mp.mpf(10)**mp.dps) # don't round
        i -= 1; A014777.pos[n] = i
    return i
A014777.pi = ''; A014777.pos = {} # M. F. Hasler, Jun 21 2022

More terms from Klaus Brockhaus, Feb 15 2007

A098266 a(1)=0; for i>=1, a(i+1)=position of first occurrence of a(i) in decimal expansion of e.

Original entry on oeis.org

0, 13, 27, 62, 32, 110, 3188, 12078, 141356, 2085932, 3497082, 4910326, 929922, 1189814, 4196683, 1301478, 19560712, 6894489, 41960008

Offset: 0

Mark Hudson (mrmarkhudson(AT)hotmail.com), Sep 01 2004

Recurrence sequence based on positions of digits in decimal places of e.

			So for example, a(2)=13 because 13th digit of e after decimal point is 0.
a(3)=27 because 27th decimal digit of e is 13, a(4)=62 because 62nd to 63rd decimal digits of e form "13" and so on.

Cf. A078197 for the first occurrence of integers in decimal digits of e; A097614 for the analogous recurrence sequence for Pi, also A014777 for positions of integers in decimal digits of Pi.

More terms from Ben Ross (bmr180(AT)psu.edu), Feb 01 2006

A103186 a(n) is the position of the start of the first occurrence of n > a(n-1) after the decimal point in Pi = 3.14159265358979323846264338327950288...

Original entry on oeis.org

1, 6, 9, 19, 31, 41, 47, 52, 55, 163, 174, 220, 281, 295, 314, 396, 428, 446, 495, 600, 650, 661, 698, 803, 822, 841, 977, 1090, 1124, 1358, 1435, 1501, 1667, 1668, 1719, 1828, 1926, 1968, 1987, 2007, 2161, 2210, 2236, 2261, 2305, 2416, 2509, 2555, 2595

Offset: 1

Suggested by Bob's Poetry Page. - Alonso del Arte, Mar 01 2005

The digits at position 1667 are "334", so according to the strict definition of this sequence, a(33) is 1667 and a(34) is 1668. However, this would not enable a person to mark in bold-face the counting numbers within the digits of pi, which was the inspiration for this sequence. Surprisingly, if overlapping is not allowed, this changes only one element of the sequence. a(34) becomes 1700 and a(35) remains 1719. No other overlapping occurs within the first 100,000 decimal digits of Pi. - Graeme McRae, Mar 20 2005

			Moving always to the right in the decimal expansion of Pi, the string "1" is found at position 1 counting from the first digit after the decimal point, the string "2" is found at position 6, the string "3" at position 6, the string "4" at position 19, etc.

Robert G. Wilson v, Table of n, a(n) for n = 1..9999
Dave Andersen, The Pi-Search Page.
Bob Happelberg, Bob's Poetry Page for Feb 2005

Cf. A000796, A078197, A014777 (another version).

k := 3000; R := RealField(k); S := IntegerToString(Round(10^k*(-3 + Pi(R)))); Q := []; d := 0; for n in [1..49] do p:= Position(S, IntegerToString(n)); d+:=p; Append(~Q, d); S := Substring(S, p+1, #S-p); end for; Q; /* Klaus Brockhaus, Feb 15 2007 */

p = ToString[ FromDigits[ RealDigits[ N[Pi - 3, 2600]][[1]]]]; lst = {0}; Do[a = StringPosition[p, ToString[n], 1][[1, 1]]; AppendTo[lst, a + lst[[ -1]]]; p = StringDrop[p, a], {n, 49}]; Rest[lst] (* Robert G. Wilson v, Mar 19 2005 *)

lista(nn, t=10^5) = {default(realprecision, t); my(d, k, v=digits(floor(Pi*10^t))); for(n=1, nn, d=digits(n); until(v[k+1..k+#d]==d, k++); print1(k, ", ")); } \\ Jinyuan Wang, Feb 18 2021

More terms from Graeme McRae and Robert G. Wilson v, Mar 19 2005

A143504 Numbers k such that k > first location of string of k in decimal expansion of e.

Original entry on oeis.org

7, 8, 18, 23, 28, 35, 36, 45, 47, 49, 52, 53, 57, 59, 60, 62, 66, 67, 69, 70, 71, 72, 74, 75, 76, 77, 81, 82, 84, 87, 90, 93, 94, 95, 96, 97, 99, 135, 138, 166, 174, 178, 181, 182, 193, 195, 200, 217, 218, 232, 233, 235, 240, 244, 247, 249, 251, 260, 264, 266

Offset: 1

Leonid Ianoushevitch (leonid163(AT)mail.ru), Oct 24 2008

'Location' starts from the first digit after the decimal point and refers to the first digit of a(n).

			1 is not a term since it is less than its location in e, 2.
7 is a term since it is greater than its location in e, 1.
18 is a term since it is greater than its location in e, 2.

Michael S. Branicky, Table of n, a(n) for n = 1..10000
Robert Nemiroff and Jerry Bonnell, The Number e to 1 Million Digits.

Cf. A001113, A038099, A078197.

from sympy import E
from itertools import count, islice
digits_of_e = str(E.n(10**5))[1:-1] # raise to 10**6 for b-file
def agen():
    for k in count(1):
        kloc = digits_of_e.find(str(k))
        assert kloc > 0, ("Increase precision", k)
        if k > kloc: yield k
print(list(islice(agen(), 60))) # Michael S. Branicky, Jul 10 2022

a(n) > A078197(n). - Michael S. Branicky, Jul 10 2022

Terms corrected and a(45) and beyond from Michael S. Branicky, Jul 10 2022

A281092 Position of the first occurrence of n in the decimal expansion of e.

Original entry on oeis.org

13, 2, 0, 17, 10, 11, 20, 1, 3, 12, 195, 200, 370, 27, 223, 201, 94, 88, 2, 108, 111, 87, 252, 16, 33, 92, 30, 0, 4, 131, 71, 189, 110, 142, 143, 17, 19, 270, 85, 106, 66, 124, 97, 134, 239, 10, 103, 25, 228, 34, 235, 93, 15, 18, 76, 301, 153, 38, 325, 11, 20, 242, 32

Offset: 0

Bobby Jacobs, Jan 21 2017

The 2 before the decimal point is counted as position 0.

This differs from A078197(n) at n = 2, 27, 271, 2718, ... .

Cf. A001113, A051238, A078197, A088576, A176341.

With[{ed=RealDigits[E,10,500][[1]]},Flatten[Table[SequencePosition[ ed, IntegerDigits[n],1][[All,1]],{n,0,65}]]]-1 (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Oct 06 2017 *)

A100080 Position of first occurrence of n after the decimal point in the decimal expansion of 1/Pi.

Original entry on oeis.org

5, 2, 26, 1, 29, 19, 9, 13, 3, 6, 297, 64, 50, 385, 45, 18, 116, 65, 2, 41, 393, 102, 85, 125, 35, 93, 26, 86, 32, 43, 4, 1, 92, 58, 59, 69, 126, 12, 165, 151, 36, 717, 437, 196, 226, 29, 60, 160, 46, 55, 30, 112, 25, 19, 108, 90, 105, 134, 123, 70, 88, 9, 446, 149, 236, 511

Offset: 0

Mark Hudson (mrmarkhudson(AT)hotmail.com), Nov 03 2004

a(0) = A133268(1),
a(1) = A134251(1),
a(2) = A134252(1),
a(3) = A134253(1),
a(4) = A134254(1),
a(5) = A134255(1),
a(6) = A134256(1),
a(7) = A134257(1),
a(8) = A134258(1),
a(9) = A134259(1),
a(10) = A134260(1). - Artur Jasinski, Oct 16 2007

			1/Pi = 0.31830988618379067153776752674... so the first occurrence of 0 after the decimal point is at position 5; first occurrence of 1 is at position 2; first occurrence of 2 is at position 26; etc.

Cf. A032445 and A014777 for Pi, A078197 for e. Digits of 1/Pi=A049541.

Cf. A037000, A014777, A133268, A134251, A134252, A134253, A134254, A134255, A134256, A134257, A134258, A134259, A134260.

Table[ SequencePosition[#, IntegerDigits@ n][[1, 1]], {n, 0, 65}] &@ First@ RealDigits@ N[1/Pi, 10^4] (* James C. McMahon, Feb 06 2024 *)

Edited by N. J. A. Sloane, Jun 30 2008 at the suggestion of R. J. Mathar

A317488 a(n) is the position of the first occurrence of n > a(n-1) after the decimal point in e = 2.71828182845904523...

Original entry on oeis.org

2, 4, 17, 25, 29, 31, 36, 86, 107, 195, 200, 370, 687, 853, 880, 899, 961, 963, 1013, 1153, 1161, 1235, 1263, 1291, 1325, 1347, 1357, 1399, 1444, 1451, 1798, 1846, 2067, 2191, 2258, 2305, 2332, 2356, 2370, 2487, 2516, 2571, 2578, 2690, 2694, 2807, 2926, 2956, 3012

Offset: 1

Philipp O. Tsvetkov, Jul 29 2018

			Moving always to the right in the decimal expansion of e, the string "1" is found at position 2 counting from the first digit after the decimal point, the string "2" is found at position 4, the string "3" at position 17, the string "4" at position 25, etc.

Cf. A001113, A078197, A103186, A281092.

p = ToString[FromDigits[RealDigits[N[E - 2, 2600]][[1]]]]; lst = {0}; Do[
a = StringPosition[p, ToString[n], 1][[1, 1]]; AppendTo[lst, a + lst[[-1]]];
p = StringDrop[p, a], {n, 29}]; Rest[lst]

A341442 a(n) is the position of the start of the first occurrence of prime(n) after the decimal point in the expansion of e.

Original entry on oeis.org

4, 17, 11, 1, 200, 27, 88, 108, 16, 131, 189, 270, 124, 134, 25, 18, 11, 242, 59, 1, 157, 168, 205, 221, 35, 195, 941, 283, 1748, 355, 370, 4604, 1574, 1998, 223, 413, 201, 483, 232, 599, 2875, 120, 1382, 108, 607, 1067, 426, 2494, 1329, 517, 178, 574, 2133

Offset: 1

Gregory Allen, Feb 11 2021

			The first position at which prime(1)=2 occurs to the right of the decimal point in e=2.71828... is the 4th digit after the decimal point, so a(1)=4.

Cf. A000040, A001113, A037024, A078197, A128050.

en=Characters[ToString@N[E,10000]];
For[x=1,x<=100,x++,Print["x=",x," ",prn=Prime[x]," ",pos=First[SequencePosition[en,Characters[ToString[prn]]]-2]]]

a(n) = A078197(prime(n)). - Rémy Sigrist, Feb 12 2021

A362058 The location of the first occurrence of n in the decimal expansion of phi (the golden ratio, 1.6180339887...).

Original entry on oeis.org

4, 0, 19, 5, 11, 22, 1, 10, 3, 7, 231, 34, 121, 55, 254, 366, 0, 35, 2, 188, 19, 54, 62, 131, 78, 213, 67, 63, 51, 174, 40, 137, 181, 5, 26, 56, 28, 98, 32, 6, 105, 90, 347, 27, 58, 21, 70, 102, 15, 11, 214, 394, 66, 111, 57, 768, 30, 48, 22, 166, 68, 1, 50

Offset: 0

James C. McMahon, Apr 06 2023

Locations in the expansion of phi are numbered 0 for the digit before the decimal point, 1 for the first digit after the decimal point, and so on.

			The first occurrence of 0 in phi occurs 4 places after the decimal point, so a(0)=4; 5 first occurs 22 places after the decimal point, so a(5)=22; 10 first occurs 231 places after the decimal point so a(10)=231.

Cf. A001622 (phi)
Cf. A088577 (1-based locations).
Cf. A078197 (for e), A176341 (for Pi), A014777 (for Pi but different indexing).

Table[-1 + SequencePosition[#, IntegerDigits@ n][[1, 1]], {n, 0, 50}] &@ First@ RealDigits@ N[GoldenRatio, 10^4]

a(n) = A088577(n) - 1.

cp's OEIS Frontend

A014777 Position of the start of the first occurrence of n after the decimal point in Pi = 3.14159265358979323846264338327950288...

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A098266 a(1)=0; for i>=1, a(i+1)=position of first occurrence of a(i) in decimal expansion of e.

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A103186 a(n) is the position of the start of the first occurrence of n > a(n-1) after the decimal point in Pi = 3.14159265358979323846264338327950288...

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A143504 Numbers k such that k > first location of string of k in decimal expansion of e.

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A281092 Position of the first occurrence of n in the decimal expansion of e.

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A100080 Position of first occurrence of n after the decimal point in the decimal expansion of 1/Pi.

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A317488 a(n) is the position of the first occurrence of n > a(n-1) after the decimal point in e = 2.71828182845904523...

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A341442 a(n) is the position of the start of the first occurrence of prime(n) after the decimal point in the expansion of e.

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