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Position of the start of the first occurrence of n after the decimal point in Pi = 3.14159265358979323846264338327950288...
+10
44
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
EXAMPLE
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.
MATHEMATICA
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 *)
PROG
(Magma) k := 700; R := RealField(k); [ Position(IntegerToString(Round(10^k*(-3 + Pi(R)))), IntegerToString(n)) : n in [0..68] ]; /* Klaus Brockhaus, Feb 15 2007 */ (PARI) 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 (Python)
from mpmath import mp
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
return i
a(1)=0; for i>=1, a(i+1)=position of first occurrence of a(i) in decimal expansion of e.
+10
14
0, 13, 27, 62, 32, 110, 3188, 12078, 141356, 2085932, 3497082, 4910326, 929922, 1189814, 4196683, 1301478, 19560712, 6894489, 41960008
COMMENTS
Recurrence sequence based on positions of digits in decimal places of e.
EXAMPLE
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.
CROSSREFS
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.
AUTHOR
Mark Hudson (mrmarkhudson(AT)hotmail.com), Sep 01 2004
EXTENSIONS
More terms from Ben Ross (bmr180(AT)psu.edu), Feb 01 2006
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...
+10
4
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
COMMENTS
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
EXAMPLE
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.
MATHEMATICA
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 *)
PROG
(Magma) 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 */ (PARI) 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
Numbers k such that k > first location of string of k in decimal expansion of e.
+10
2
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
COMMENTS
'Location' starts from the first digit after the decimal point and refers to the first digit of a(n).
EXAMPLE
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.
PROG
(Python)
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
AUTHOR
Leonid Ianoushevitch (leonid163(AT)mail.ru), Oct 24 2008
Position of the first occurrence of n in the decimal expansion of e.
+10
1
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
COMMENTS
The 2 before the decimal point is counted as position 0.
This differs from A078197(n) at n = 2, 27, 271, 2718, ... .
MATHEMATICA
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 *)
Position of first occurrence of n after the decimal point in the decimal expansion of 1/Pi.
+10
0
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
EXAMPLE
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.
MATHEMATICA
Table[ SequencePosition[#, IntegerDigits@ n][[1, 1]], {n, 0, 65}] &@ First@ RealDigits@ N[1/Pi, 10^4] (* James C. McMahon, Feb 06 2024 *)
CROSSREFS
Cf. A037000, A014777, A133268, A134251, A134252, A134253, A134254, A134255, A134256, A134257, A134258, A134259, A134260.
AUTHOR
Mark Hudson (mrmarkhudson(AT)hotmail.com), Nov 03 2004
a(n) is the position of the first occurrence of n > a(n-1) after the decimal point in e = 2.71828182845904523...
+10
0
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
EXAMPLE
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.
MATHEMATICA
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]
a(n) is the position of the start of the first occurrence of prime(n) after the decimal point in the expansion of e.
+10
0
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
EXAMPLE
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.
MATHEMATICA
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]]]
The location of the first occurrence of n in the decimal expansion of phi (the golden ratio, 1.6180339887...).
+10
0
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
COMMENTS
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.
EXAMPLE
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.
MATHEMATICA
Table[-1 + SequencePosition[#, IntegerDigits@ n][[1, 1]], {n, 0, 50}] &@ First@ RealDigits@ N[GoldenRatio, 10^4]
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