A109514 -id:A109514 - cp's OEIS frontend

A057679 Self-locating strings within Pi: numbers n such that the string n is at position n in the decimal digits of Pi, where 3 is the first digit.

5, 242424, 271070, 9292071, 29133316, 70421305, 215817165252, 649661007154

Offset: 1

Mike Keith, Oct 19 2000

The average number of matches of length "n" digits is exactly 0.9. That is, we expect 0.9 matches with 1 digit, 0.9 matches with 2 digits, etc. Increasing the number of digits by a factor of 10 means that we expect to find 0.9 new matches. Increasing the search from 10^11 to 10^12 (which includes 10 times as much work) would thus only expect to find 0.9 new matches. - Alan Eliasen, May 01 2013 (corrected by Michael Beight, Mar 21 2020)
a(2) is not the first occurrence of 242424 in Pi (which is at position 242422) but the second. - Hans Havermann, Jul 26 2014
a(9) is greater than 5 * 10^13. - Kang Seonghoon, Nov 02 2020

			5 is a term because 5 is the 5th digit of Pi (3.1415...).

Tom Crawford and Brady Haran, Strings and Loops within Pi, Numberphile video (2020).
Google, 50 trillion digits of pi (2020).

Cf. A000796, A057680, A064810, A109514.

StringsinPi[m_] := Module[{cc = 10^m + m, sol, aa}, sol = Partition[RealDigits[Pi,10,cc] // First, m, 1]; Do[aa = FromDigits[sol[[i]]]; If[aa==i, Print[{i, aa}]], {i,Length[sol]}];] (* For example, StringsinPi[6] returns all 6-digit members of the sequence. - Colin Rose, Mar 15 2006 *)
dpi = RealDigits[Pi, 10, 10000010][[1]]; Select[Range[10000000], FromDigits[Take[dpi, {#, # - 1 + IntegerLength[#]}]] == # &] (* Vaclav Kotesovec, Feb 18 2020 *)

a(4)-a(6) from Colin Rose, Mar 15 2006
a(7) from Alan Eliasen, May 10 2013
a(8) from Alan Eliasen, Jun 06 2013
Name clarified by Kang Seonghoon, Nov 02 2020

A109513 Let k be an m-digit integer. Then k is a Pithy number if the k-th m-tuple in the decimal digits of Pi (after the decimal point) is the string k.

Original entry on oeis.org

1, 19, 94, 3542, 7295, 318320, 927130, 939240, 688370303, 7682437410, 7996237896, 89594051933

Offset: 0

Colin Rose, Jul 01 2005

			1 is a term because the first digit in Pi (after the decimal point) is 1.
19 is a term because the 19th pair of digits (after the decimal point) in Pi is 19:
      1  2  3  4  5  6  7  8  9 10 11 12 13 14 15 16 17 18 19
  3. 14 15 92 65 35 89 79 32 38 46 26 43 38 32 79 50 28 84 19 ...

David G. Andersen, The Pi-Search Page.

Cf. A000796, A109514, A057679, A057680.

PithyNumbers[m_] := Module[{cc = m(10^m)+m, sol, aa}, sol = Partition[RealDigits[Pi, 10, cc] // First // Rest, m]; Do[aa = FromDigits[sol[[i]]]; If[aa==i, Print[{i, aa}]], {i, Length[sol]}];] Example: PithyNumbers[4] produces all 4-digit Pithy numbers

a(8)-a(11) from J. Volkmar Schmidt, Dec 17 2023

A153221 Numbers k such that the string k is found at position k-3 in the decimal digits of Pi.

Original entry on oeis.org

51, 875, 62843, 242424, 4308765, 9710721, 24747689, 126987778

Offset: 1

Gil Broussard, Dec 21 2008

			a(1) = 51 because 51 occurs at offset (51-3) after the decimal in the digits of Pi.

Cf. A057679, A057680, A109513, A109514, A153223, A153224.

from sympy import S
# download https://stuff.mit.edu/afs/sipb/contrib/pi/pi-billion.txt, then
#with open('pi-billion.txt', 'r') as f: pi_digits = f.readline()
pi_digits = str(S.Pi.n(3*10**5+2))[:-2] # alternative to above
pi_digits = pi_digits.replace(".", "")
def ispal(s): return s == s[::-1]
def afind():
    for k in range(len(pi_digits)):
        sk = str(k)
        if sk == pi_digits[k-3:k-3+len(sk)]:
            print(k, end=", ")
afind() # Michael S. Branicky, Jan 30 2022

a(5)-a(8) from Michael S. Branicky, Jan 30 2022

A153223 Numbers k such that the string k is found at position k-4 in the decimal digits of Pi.

Original entry on oeis.org

9, 233, 1614, 9218, 27755974, 81259258, 120526011, 238732548, 651265325, 783609697

Offset: 1

Gil Broussard, Dec 21 2008

			a(1) = 9 because 9 occurs at offset (9-4) after the decimal in the digits of Pi.

Cf. A057679, A057680, A109513, A109514, A153221, A153224.

from sympy import S
# download https://stuff.mit.edu/afs/sipb/contrib/pi/pi-billion.txt, then
#with open('pi-billion.txt', 'r') as f: pi_digits = f.readline()
pi_digits = str(S.Pi.n(3*10**5+2))[:-2] # alternative to above
pi_digits = pi_digits.replace(".", "")
def afind():
    for k in range(5, len(pi_digits)):
        sk = str(k)
        if sk == pi_digits[k-4:k-4+len(sk)]:
            print(k, end=", ")
afind() # Michael S. Branicky, Jan 30 2022

a(5)-a(10) from Michael S. Branicky, Jan 30 2022

A153224 Numbers k such that the string k is found at position k-5 in the decimal digits of Pi.

Original entry on oeis.org

26, 32, 41, 86, 2799, 469113068

Offset: 1

Gil Broussard, Dec 21 2008

			a(1) = 26 because 26 occurs at offset (26-5) after the decimal in the digits of Pi.

Cf. A057679, A057680, A109513, A109514, A153221, A153223.

from sympy import S
# download https://stuff.mit.edu/afs/sipb/contrib/pi/pi-billion.txt, then
#with open('pi-billion.txt', 'r') as f: pi_digits = f.readline()
pi_digits = str(S.Pi.n(3*10**5+2))[:-2] # alternative to above
pi_digits = pi_digits.replace(".", "")
def afind():
    for k in range(6, len(pi_digits)):
        sk = str(k)
        if sk == pi_digits[k-5:k-5+len(sk)]:
            print(k, end=", ")
afind() # Michael S. Branicky, Jan 30 2022

a(6) from Michael S. Branicky, Jan 30 2022

A117432 Let n be an integer consisting of m digits. Then n is a Phithy number if the n-th m-tuple in the decimal digits of golden ratio phi is string n.

Original entry on oeis.org

1, 20, 63, 104, 7499430, 9228401

Offset: 0

Colin Rose, Mar 14 2006

			1 is a term because the first single digit in golden ratio phi is 1.
Number 20 is a term because the 20th pair of digits in phi is 20.
(cf. phi = 1.6180339887498948482045868343656381177203...)

Eric Weisstein's World of Mathematics, The Golden Ratio

Cf. A001622, A109513, A109514, A117431.

PhithyNumbers[m_] := Module[{cc = m(10^m)+m, sol, aa}, sol = Partition[RealDigits[GoldenRatio, 10, cc] // First, m]; Do[aa = FromDigits[sol[[i]]]; If[aa==i, Print[{i, aa}]], {i,Length[sol]}];] Example: PhithyNumbers[3] produces all 3-digit Phithy numbers

from sympy import S
def aupto(nn):
  mm = len(str(nn))
  phistr = str(S.GoldenRatio.n(nn*mm+1)).replace(".", "")[:-1]
  for n in range(1, nn+1):
    nstr = str(n)
    m = len(nstr)
    if phistr[(n-1)*m:n*m] == nstr: print(n, end=", ")
aupto(10**5) # Michael S. Branicky, Jan 20 2021

a(4)-a(5) from Michael S. Branicky, Jan 21 2021

A153220 Self-locating strings within Pi: numbers n such that the string n is at position n (counting 3 and the decimal point) in decimal digits of Pi.

Original entry on oeis.org

4, 315, 360, 384, 47696, 496498, 1577526, 5607861220, 6106488294

Offset: 1

Gil Broussard, Dec 21 2008

a(8) > 10^9. - Vaclav Kotesovec, Feb 18 2020

			a(1)=4 because the 4th character (including the decimal point) in 3.14159... is also a 4.

Cf. A057679, A057680, A109513, A109514

dpi = RealDigits[Pi, 10, 10000010][[1]]; Select[Range[2, 10000000], FromDigits[Take[dpi, {# - 1, # - 2 + IntegerLength[#]}]] == # &] (* Vaclav Kotesovec, Feb 18 2020 *)

a(7) from Vaclav Kotesovec, Feb 18 2020

a(8)-a(9) from Guixin Lai, Aug 22 2025

A153225 Numbers k such that the string k modulo 100 is found at position k in the decimal digits of Pi.

Original entry on oeis.org

1, 102, 104, 189, 193, 256, 302, 407, 467, 475, 503, 594, 702, 712, 751, 804, 881, 905, 978, 998, 1005, 1053, 1104, 1107, 1154, 1275, 1303, 1306, 1307, 1315, 1421, 1502, 1600, 1604, 1690, 1694, 1706, 1802, 1860, 1904, 1907, 1908, 2006, 2025, 2105, 2146, 2208

Offset: 1

Gil Broussard, Dec 21 2008

			a(4) = 189 because 89 occurs at offset 189 after the decimal in the digits of Pi.

Michael S. Branicky, Table of n, a(n) for n = 1..10000

Cf. A000796, A057679, A057680, A109513, A109514.

from sympy import S
# download https://stuff.mit.edu/afs/sipb/contrib/pi/pi-billion.txt, then
#with open('pi-billion.txt', 'r') as f: pi_digits = f.readline()
pi_digits = str(S.Pi.n(3*10**5+2))[:-2] # alternative to above
pi_digits = pi_digits.replace(".", "")
def ispal(s): return s == s[::-1]
def agen():
    for k in range(len(pi_digits)):
        sk = str(k%100)
        if sk == pi_digits[k:k+len(sk)]:
            yield k
g = agen()
print([next(g) for n in range(1, 48)]) # Michael S. Branicky, Jan 30 2022

a(47) and beyond from Michael S. Branicky, Jan 30 2022

A153226 Numbers k such that the string k modulo 1000 is found at position k in the decimal digits of Pi.

Original entry on oeis.org

1, 1005, 1053, 1255, 2006, 2025, 2246, 2560, 3712, 4063, 4066, 4087, 5006, 5009, 5038, 5068, 5076, 5538, 6000, 6025, 6045, 7007, 7025, 7037, 8068, 8960, 9009, 9052, 10007, 10823, 11003, 11005, 12000, 12003, 12134, 12639, 14009, 14207, 14326, 14944, 15052, 16000

Offset: 1

Gil Broussard, Dec 21 2008

			a(4) = 1255 because 255 occurs at offset 1255 after the decimal in the digits of Pi.

Michael S. Branicky, Table of n, a(n) for n = 1..10000

Cf. A000796, A057679, A057680, A109513, A109514.

from sympy import S
# download https://stuff.mit.edu/afs/sipb/contrib/pi/pi-billion.txt, then
#with open('pi-billion.txt', 'r') as f: pi_digits = f.readline()
pi_digits = str(S.Pi.n(3*10**5+2))[:-2] # alternative to above
pi_digits = pi_digits.replace(".", "")
def ispal(s): return s == s[::-1]
def agen():
    for k in range(len(pi_digits)):
        sk = str(k%1000)
        if sk == pi_digits[k:k+len(sk)]:
            yield k
g = agen()
print([next(g) for n in range(1, 43)]) # Michael S. Branicky, Jan 30 2022

a(40) and beyond from Michael S. Branicky, Jan 30 2022

A153227 Numbers k such that the string k is found at position 2k in the decimal digits of Pi.

Original entry on oeis.org

5, 95, 171, 529, 1913, 2753, 60597, 831092, 9552919

Offset: 1

Gil Broussard, Dec 21 2008

			a(2) = 95 because 95 occurs at offset 190 (2*95) after the decimal in the digits of Pi.

Cf. A057679, A057680, A109513, A109514, A153228.

from sympy import S
# download https://stuff.mit.edu/afs/sipb/contrib/pi/pi-billion.txt, then
#with open('pi-billion.txt', 'r') as f: pi_digits = f.readline()
pi_digits = str(S.Pi.n(3*10**5+2))[:-2] # alternative to above
pi_digits = pi_digits.replace(".", "")
def afind():
    for k in range(1, len(pi_digits)):
        sk = str(k)
        if sk == pi_digits[2*k:2*k+len(sk)]:
            print(k, end=", ")
afind() # Michael S. Branicky, Jan 30 2022

a(8) corrected by Paul Tek, Jun 09 2013

a(9) from Michael S. Branicky, Jan 30 2022

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A057679 Self-locating strings within Pi: numbers n such that the string n is at position n in the decimal digits of Pi, where 3 is the first digit.

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A109513 Let k be an m-digit integer. Then k is a Pithy number if the k-th m-tuple in the decimal digits of Pi (after the decimal point) is the string k.

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A153221 Numbers k such that the string k is found at position k-3 in the decimal digits of Pi.

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A153223 Numbers k such that the string k is found at position k-4 in the decimal digits of Pi.

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A153224 Numbers k such that the string k is found at position k-5 in the decimal digits of Pi.

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A117432 Let n be an integer consisting of m digits. Then n is a Phithy number if the n-th m-tuple in the decimal digits of golden ratio phi is string n.

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A153220 Self-locating strings within Pi: numbers n such that the string n is at position n (counting 3 and the decimal point) in decimal digits of Pi.

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Mathematica

Extensions

A153225 Numbers k such that the string k modulo 100 is found at position k in the decimal digits of Pi.

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Python

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