A057679 -id:A057679 - cp's OEIS frontend

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

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

A153228 Numbers k such that the string k is found at position 3k in the decimal digits of Pi.

Original entry on oeis.org

1, 2, 3, 85, 200, 447263

Offset: 1

Gil Broussard, Dec 21 2008

			a(4) = 85 because 85 occurs at offset 255 (3*85) after the decimal in the digits of Pi.

Cf. A057679, A057680, A109513, A109514, A153227.

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[3*k:3*k+len(sk)]:
            print(k, end=", ")
afind() # Michael S. Branicky, Jan 30 2022

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

A366830 Self-locating strings within Pi: numbers n such that the reverse string n starts at position n in the decimal digits of Pi, where 3 is the first digit.

Original entry on oeis.org

5, 1610, 5833, 82699856, 9633661255, 17292288245, 78246420246

Offset: 1

J. Volkmar Schmidt, Oct 31 2023

			1610 is a term because '0161' starts at position 1610:
Position  1  2  3  4  ...  1610  1611  1612  1613  ...
Pi        3  1  4  1  ...    0     1     6     1   ...

Cf. A057679, A057680, A366831.

A366831 Self-locating strings within Pi: numbers n such that the reverse string n starts at position n in the decimal digits of Pi, where leading 3 is omitted.

Original entry on oeis.org

1, 50, 576, 242424, 746074, 10311073, 54848721, 103849853, 48438192357

Offset: 1

J. Volkmar Schmidt, Oct 31 2023

			50 is a term because '05' starts at position 50,
576 is a term because '675' starts at position 576:
Position        1 2 3 ... 50 51 ... 576 577 578 ...
Pi (3 omitted)  1 4 1 ... 0  5  ... 6   7   5   ...

Cf. A057679, A057680, A366830.

A153230 Numbers k such that the string k is found at position k^2 in the decimal digits of Pi.

Original entry on oeis.org

1, 3, 22, 596, 43934, 78778, 249994

Offset: 1

Gil Broussard, Dec 21 2008

a(5) > 10^4, if it exists. - Amiram Eldar, May 06 2024
From Michael S. Branicky, May 19 2025: (Start)
a(8) > 10^6.
Assuming uniformly random digits, the expected value is 0.9 terms per digit-length. (End)

			22 is in the sequence because 22 occurs at position 484 (22^2) after the decimal point in the digits of Pi.

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

With[{m = 10^3}, s = Rest@ RealDigits[Pi, 10, m^2][[1]]; q[n_] := s[[n^2 + Range[0, IntegerLength[n] - 1]]] == RealDigits[n][[1]]; Select[Range[m - 1], q]] (* Amiram Eldar, May 06 2024 *)

a(5)-a(7) from Michael S. Branicky, May 18 2025

A331015 Self-locating strings within Euler-Mascheroni Constant (gamma), strings k at position k (taking gamma offset 1).

Original entry on oeis.org

57, 16939, 767158, 5505709, 6674196, 7418985, 18873720

Offset: 1

S. Alwin Mao, Feb 12 2020

The first self-locating digits of gamma are 57, and the first digits of gamma are 57.

A near-miss '04305165' begins at position 4305165.

			57 is a term because the 57th digit is 5 and the 58th digit is 7.

Euler-Mascheroni constant digits: A001620.

Self-locating digits of Pi: A057679, A064810 and e: A205648.

dgamma = RealDigits[EulerGamma, 10, 1000010][[1]]; Select[Range[1000000], FromDigits[Take[dgamma, {#, # - 1 + IntegerLength[#]}]] == # &] (* Vaclav Kotesovec, Feb 18 2020 *)

A332492 Self-locating numbers within the decimal expansion of log(2): strings k beginning at position k (first digit after decimal point is position 2).

Original entry on oeis.org

7, 23, 52, 54, 86, 303, 389, 5112, 60392, 87491, 94788, 97115, 616916, 672938066

Offset: 1

S. Alwin Mao, Feb 14 2020

a(15) > 10^9.

52 and 54 are self-locating numbers which appear consecutively in log(2) = 0.693...5254....

			For 0.693147..., if 6 appears at position 2, then 7 appears at position 7.

Decimal expansion of log(2): A002162.

Self-locating digits of Pi: A057679, A153220.

dlog2 = RealDigits[Log[2], 10, 1000010][[1]]; Select[Range[2, 1000000], FromDigits[Take[dlog2, {# - 1, # - 2 + IntegerLength[#]}]] == # &] (* Vaclav Kotesovec, Feb 18 2020 *)

cp's OEIS Frontend

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

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Python

Extensions

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

Views

Author

Keywords

Examples

Crossrefs

Programs

Python

Extensions

A153228 Numbers k such that the string k is found at position 3k in the decimal digits of Pi.

Views

Author

Keywords

Examples

Crossrefs

Programs

Python

Extensions

A366830 Self-locating strings within Pi: numbers n such that the reverse string n starts at position n in the decimal digits of Pi, where 3 is the first digit.

Views

Author

Keywords

Examples

Crossrefs

A366831 Self-locating strings within Pi: numbers n such that the reverse string n starts at position n in the decimal digits of Pi, where leading 3 is omitted.

Views

Author

Keywords

Examples

Crossrefs

A153230 Numbers k such that the string k is found at position k^2 in the decimal digits of Pi.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Extensions

A331015 Self-locating strings within Euler-Mascheroni Constant (gamma), strings k at position k (taking gamma offset 1).

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A332492 Self-locating numbers within the decimal expansion of log(2): strings k beginning at position k (first digit after decimal point is position 2).

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica