Mike Keith - cp's OEIS frontend

A374380 Numbers b such that the base-b expansion of Pi has two equal digits immediately after the radix point.

2, 8, 114, 227, 340, 453, 566, 679, 33103, 66318, 265382, 1360121, 1725034, 25510583, 78256780, 209259756, 340262732, 1151791170, 1963319608, 6701487260, 13402974519, 20104461778, 26805949037, 33507436296, 40208923555, 46910410814, 53611898073, 574364584668

Offset: 1

Mike Keith, Jul 06 2024

nonn

The probability that this occurs in a random base-b number is 1/b, so the divergence of the harmonic series suggests this sequence is infinite. a(28) = 574364584668 is the largest term < 10^12.

If 1 is added to terms of A002486, many of the resulting numbers (roughly 60%) are terms of this sequence. - Mike Keith, Jul 28 2024

			2 is a term since Pi in base 2 is 11.00... which has two equal digits 0,0.
114 is a term since Pi in base 114 is 3.(16)(16)... which has two equal "digits" 16 and 16.

Cf. A000796 (Pi), A336017 (second digit after radix point), A002486 (continued fraction convergent denominators).

A134996 Dihedral calculator primes: p, p upside down, p in a mirror, p upside-down-and-in-a-mirror are all primes.

Original entry on oeis.org

2, 5, 11, 101, 181, 1181, 1811, 18181, 108881, 110881, 118081, 120121, 121021, 121151, 150151, 151051, 151121, 180181, 180811, 181081, 188011, 188801, 1008001, 1022201, 1028011, 1055501, 1058011, 1082801, 1085801, 1088081, 1108201, 1108501, 1110881, 1120121, 1120211

Offset: 1

Mike Keith

The digits of a(n) are restricted to 0, 1, 2, 5, 8. - Ivan N. Ianakiev, Oct 08 2015

The first term containing all the possible digits is 108225151. There are 2958 such terms up to 10^12, the last one in this range being 188885250551. - Giovanni Resta, Oct 08 2015

			120121 is such a number because 120121, 121021 (upside down), 151051 (mirror) and 150151 are all prime. (This is the smallest one in which all four numbers are distinct.)

Alois P. Heinz, Table of n, a(n) for n = 1..7174
C. K. Caldwell, The Prime Glossary, Dihedral Prime
Eric Weisstein's World of Mathematics, Dihedral Prime.

Cf. A134997, A134998, A048661.

lst1={2,5};
startQ[n_]:=First[IntegerDigits[n]]==1;
subQ[n_]:=Module[{lst={0,1,2,5,8}},SubsetQ[lst,Union[IntegerDigits[n]]]];
rev[n_]:=Reverse[IntegerDigits[n]];
updown[n_]:=FromDigits[rev[n]];
mirror[n_]:=FromDigits[rev[n]/.{2-> 5,5-> 2}];
updownmirror[n_]:=FromDigits[rev[mirror[n]]];
lst2=Select[Range@188801,And[startQ[#],subQ[#],PrimeQ[#],PrimeQ[updown[#]],PrimeQ[mirror[#]],PrimeQ[updownmirror[#]]]&];
Join[lst1,lst2] (* Ivan N. Ianakiev, Oct 08 2015 *)

from sympy import isprime
from itertools import count, islice, product
def t(s): return s.translate({ord("2"):ord("5"), ord("5"):ord("2")})
def ok(s): # s is a string of digits
    return all(isprime(int(w)) for w in [s, s[::-1], t(s), t(s[::-1])])
def agen(): # generator of terms
    yield from (2, 5)
    for d in count(2):
        for mid in product("01258", repeat=d-2):
            s = "1" + "".join(mid) + "1"
            if ok(s): yield int(s)
print(list(islice(agen(), 35))) # Michael S. Branicky, Apr 27 2024

5 added by Patrick Capelle, Feb 06 2008

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.

Original entry on oeis.org

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

A058760 Integers whose set of prime factors (taken with multiplicity) uses each digit exactly once (i.e., is pandigital) in some base b > 1. Numbers are expressed in base 10.

Original entry on oeis.org

2, 6, 11, 19, 38, 174, 190, 230, 365, 377, 417, 445, 485, 493, 537, 553, 565, 606, 681, 721, 813, 869, 1133, 1313, 1717, 2866, 3946, 4546, 5631, 5662, 5811, 6322, 6351, 6711, 6730, 6855, 7971

Offset: 0

G. L. Honaker, Jr. and Mike Keith, Jan 01 2001

Note that the appropriate base for each number is not given here.

			19 in base 3 is 201, which is pandigital, so it qualifies (having just itself as a prime factor). A more elaborate example is 5662 = 2 * 19 * 149, which in base 6 is 2 * 31 * 405, also pandigital.

A058909 Integers whose set of prime factors (taken with multiplicity) uses each digit exactly once (i.e., is pandigital) in base 10.

Original entry on oeis.org

15618090, 22022490, 22816290, 22908090, 23294190, 23507490, 26679990, 27114690, 27687090, 28275690, 29447898, 29544690, 29582490, 29670378, 29910138, 30134238, 30426918, 31207890, 31406910, 31430670, 31490610, 32024670, 32035470, 32054910

Offset: 1

Mike Keith and G. L. Honaker, Jr., Jan 08 2001

The last term is a(248769) = 8439563243 = 9643*875201. - Giovanni Resta, Nov 20 2019

			15618090 is in the sequence since 2 * 3 * 5 * 487 * 1069 is pandigital; 29447898 because 2 * 3 * 107 * 45869 is pandigital.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A058760.

A057680 Self-locating strings within Pi: numbers n such that the string n is at position n in the decimal digits of Pi, where the initial digit 3 is at position 0.

Original entry on oeis.org

1, 16470, 44899, 79873884, 711939213, 36541622473, 45677255610, 62644957128, 656430109694

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)
Consequently, with the second Borel-Cantelli lemma, the expected number of terms in this sequence is infinite with probability 1. (Of course the sequence is not random, but almost all of the sequences corresponding to randomly-chosen real numbers in place of Pi have infinitely many terms.) - Charles R Greathouse IV, Apr 29 2013
a(1) & a(5) are the first occurrences in Pi of their respective strings; a(2) & a(4) are the second occurrences; a(3) is the fourth occurrence. - Hans Havermann, Jul 27 2014
A near-miss '043611' occurs at position 43611. - S. Alwin Mao, Feb 18 2020
a(10) > 5 * 10^13. - Kang Seonghoon, Nov 02 2020
Has no terms in common with A037008 (but see Mao comment above). - Charles R Greathouse IV, Jun 21 2022

			1 is a term because the string of digits '1' occurs as the 1st digit after the decimal point.
Similarly, 16470 is a term because the string of digits '16470' occurs starting at position 16470 (after the decimal point) in the digits of Pi (although it already occurs earlier at position 1602). - _M. F. Hasler_, Jul 29 2024

Clifford A. Pickover, A Passion for Mathematics, Wiley, 2005; see p. 60.

David G. Andersen, The Pi-Search Page.
Tom Crawford and Brady Haran, Strings and Loops within Pi, Numberphile video (2020).
Google, 50 trillion digits of pi (2020).

Cf. A000796 (decimal digits of Pi), A057679 (variant where position 1 refers to the initial digit 3), A064810 (variant where position 0 refers to the first digit after the decimal point), A109513 (variant using chunks of m digits).

StringsinPiAfterPoint[m_] := Module[{cc = 10^m + m, sol, aa}, sol = Partition[RealDigits[Pi,10,cc] // First // Rest, m, 1]; Do[aa = FromDigits[sol[[i]]]; If[aa==i, Print[{i, aa}]], {i,Length[sol]}];] (* For example, StringsinPiAfterPoint[5] returns all 5-digit members of the sequence. - Colin Rose, Mar 15 2006 *)
Do[If[RealDigits[Pi,10,a=i+IntegerLength@i-1,-1][[1,i;;a]]==IntegerDigits@i,Print@i],{i,50000}] (* Giorgos Kalogeropoulos, Feb 21 2020 *)

A057680_row(r=5)={my(M=10^r, R=[]); for(n=M\10, M-1, localprec(n+r); Pi\10^(1-r-n)%M==n && !print1(n",") && R=concat(R,n));R} \\ prints & returns the r-digit terms. - M. F. Hasler, Jul 29 2024

More terms from Colin Rose, Mar 15 2006
a(5) from Nathaniel Johnston, Nov 12 2010
a(6)-a(8) from Alan Eliasen, May 01 2013
a(9) from Alan Eliasen, Jun 06 2013
Name clarified by Kang Seonghoon, Nov 02 2020
Edited by M. F. Hasler, Jul 29 2024

A055055 Number of basic magic carpets on n points.

Original entry on oeis.org

1, 10, 271, 36995

Offset: 5

Mike Keith, Jun 02 2000

E. Friedman and M. Keith, Magic Carpets, J. Int Sequences, 3 (2000), #P.00.2.5.

Typo in a(8) corrected by Michel Marcus, Oct 07 2016

A055056 Number of smooth basic magic carpets on n points.

Original entry on oeis.org

1, 1, 7, 55, 1306

Offset: 5

Mike Keith, Jun 02 2000

E. Friedman and M. Keith, Magic Carpets, J. Int Sequences, 3 (2000), #P.00.2.5.

A055057 Largest k value that admits an (n,k) smooth magic carpet.

Original entry on oeis.org

3, 3, 8, 14, 15

Offset: 5

Mike Keith, Jun 02 2000

E. Friedman and M. Keith, Magic Carpets, J. Int Sequences, 3 (2000), #P.00.2.5.

A055605 Number of balanced basic magic carpets on n points.

Original entry on oeis.org

1, 1, 38, 2324

Offset: 5

Mike Keith, Jun 02 2000

E. Friedman and M. Keith, Magic Carpets, J. Int Sequences, 3 (2000), #P.00.2.5.

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User: Mike Keith

A374380 Numbers b such that the base-b expansion of Pi has two equal digits immediately after the radix point.

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A134996 Dihedral calculator primes: p, p upside down, p in a mirror, p upside-down-and-in-a-mirror are all primes.

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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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A058760 Integers whose set of prime factors (taken with multiplicity) uses each digit exactly once (i.e., is pandigital) in some base b > 1. Numbers are expressed in base 10.

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A058909 Integers whose set of prime factors (taken with multiplicity) uses each digit exactly once (i.e., is pandigital) in base 10.

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A057680 Self-locating strings within Pi: numbers n such that the string n is at position n in the decimal digits of Pi, where the initial digit 3 is at position 0.

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A055055 Number of basic magic carpets on n points.

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A055056 Number of smooth basic magic carpets on n points.

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A055057 Largest k value that admits an (n,k) smooth magic carpet.

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A055605 Number of balanced basic magic carpets on n points.

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