A103186 -id:A103186 - 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

A307463 a(n) is the digit after the appearance of n in the decimal numbers of Pi after all the previous natural numbers of n have already appeared except for 0, and without overlap.

Original entry on oeis.org

4, 6, 5, 6, 0, 9, 5, 2, 7, 2, 0, 8, 3, 1, 5, 0, 3, 5, 4, 0, 4, 4, 5, 4, 2, 1, 8, 3, 5, 0, 5, 1, 4, 5, 6, 0, 1, 9, 0, 7, 0, 7, 1, 6, 1, 7, 9, 9, 3, 6, 2, 0, 9, 8, 2, 4, 0, 6, 8, 8, 4, 2, 4, 7, 4, 6, 2, 7, 5, 1, 6, 0, 4, 1, 4, 3, 6, 3, 6, 7, 6, 9, 1, 5, 3, 0, 4, 1

Offset: 1

Marc Bofill Janer, Apr 09 2019

All terms appear in the decimal expansion of Pi.

			See the first decimal digits of Pi for the examples:
3.(1)4159(2)65(3)589793238(4)62643383279(5)0288...
In parentheses the first appearing of natural numbers after all smaller natural numbers have already appeared.
- FIRST ELEMENT a(1):
For n=1, the first '1' appears in the first decimal place of Pi, and the next decimal digit is '4', so a(1)=4.
- DIGIT POSITION:
For n=4, although the first 4 appears in the 2 decimal place, not all the previous natural numbers of 4 have appeared, so, after 1, 2, and 3 have appeared (in this order), then, a(4) will be the next digit after the next 4. So a(4)=6.
- N WITH MORE THAN 1 DIGIT:
In the decimal digits of Pi: ...50284(10)270193852(11)05...
For n with more than 1 digit, a(n) is, after all the previous natural numbers have appeared, the next digit after all the digits of n have appeared consecutively. Example: a(10)=2, a(11)=0.
- NO OVERLAP:
In the decimal digits of Pi: ...52230825(33)44685035...730359825(34)904...
Example: for n=33, a(33)=4, but, as there is no overlap, the '3' cannot be used again with the '4' for n=34, so a(34) is defined by the next 34: a(34)=9.

Cf. A000796, A103186.

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

More terms from Jinyuan Wang, Feb 18 2021

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]

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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A307463 a(n) is the digit after the appearance of n in the decimal numbers of Pi after all the previous natural numbers of n have already appeared except for 0, and without overlap.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

PARI

Extensions

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

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica