cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-6 of 6 results.

A018247 The 10-adic integer x = ...8212890625 satisfying x^2 = x.

Original entry on oeis.org

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

Views

Author

Yoshihide Tamori (yo(AT)salk.edu)

Keywords

Comments

The 10-adic numbers a and b defined in this sequence and A018248 satisfy a^2=a, b^2=b, a+b=1, ab=0. - Michael Somos

Examples

			x = ...0863811000557423423230896109004106619977392256259918212890625.

References

  • W. W. R. Ball, Mathematical Recreations & Essays, N.Y. Macmillan Co, 1947.
  • V. deGuerre and R. A. Fairbairn, Jnl. Rec. Math., No. 3, (1968), 173-179.
  • M. Kraitchik, Sphinx, 1935, p. 1.

Links

Crossrefs

A007185 gives associated automorphic numbers.
Cf. A018248, A033819, A003226.
The difference between A018248 & this sequence is A075693 and their product is A075693.
The six examples given by deGuerre and Fairbairn are A055620, A054869, A018247, A018248, A259468, A259469.

Programs

  • Mathematica
    a = {5}; f[n_] := Block[{k = 0, c}, While[c = FromDigits[Prepend[a, k]]; Mod[c^2, 10^n] != c, k++ ]; a = Prepend[a, k]]; Do[ f[n], {n, 2, 105}]; Reverse[a]
With[{n = 150}, Reverse[IntegerDigits[PowerMod[5, 2^n, 10^n]]]] (* IWABUCHI Yu(u)ki, Feb 16 2024 *)
  • PARI
    a(n)=local(t=5);for(k=1,n+1,t=t^2%10^k);t\10^n \\ Paul D. Hanna, Jul 08 2006
  • PARI
    Vecrev(digits(lift(chinese(Mod(1, 2^100), Mod(0, 5^100))))) \\ Seiichi Manyama, Aug 07 2019

Formula

x = 10-adic lim_{n->oo} 5^(2^n) mod 10^(n+1). - Paul D. Hanna, Jul 08 2006

Extensions

More terms from David W. Wilson
Edited by David W. Wilson, Sep 26 2002

A018248 The 10-adic integer x = ...1787109376 satisfies x^2 = x.

Original entry on oeis.org

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

Views

Author

Yoshihide Tamori (yo(AT)salk.edu)

Keywords

Comments

The 10-adic numbers a and b defined in A018247 and this sequence satisfy a^2=a, b^2=b, a+b=1, ab=0. - Michael Somos

Examples

			x equals the limit of the (n+1) trailing digits of 6^(5^n):
6^(5^0)=(6), 6^(5^1)=77(76), 6^(5^2)=28430288029929701(376), ...
x = ...9442576576769103890995893380022607743740081787109376.
From _Peter Bala_, Nov 05 2022: (Start)
Trailing digits of 2^(10^n), 4^(10^n) and 6^(10^n) for n = 5:
2^(10^5) = ...9883(109376);
4^(10^5) = ...7979(109376);
6^(10^5) = ...4155(109376). (End)

References

  • W. W. R. Ball, Mathematical Recreations & Essays, N.Y. Macmillan Co, 1947.
  • R. Cuculière, Jeux Mathématiques, in Pour la Science, No. 6 (1986), 10-15.
  • V. deGuerre and R. A. Fairbairn, Automorphic numbers, J. Rec. Math., 1 (No. 3, 1968), 173-179.
  • M. Kraitchik, Sphinx, 1935, p. 1.
  • A. M. Robert, A Course in p-adic Analysis, Springer, 2000; see pp. 63, 419.

Links

Crossrefs

A016090 gives associated automorphic numbers.
Cf. A018247, A033819.
The difference between this sequence & A018247 is A075693 and their product is A075693.
Cf. A120817, A120818, A091664.
The six examples given by deGuerre and Fairbairn are A055620, A054869, A018247, A018248, A259468, A259469.

Programs

  • Maple
    a := proc (n) option remember; if n = 1 then 2 else irem(a(n-1)^10, 10^n) end if; end proc:
# display the digits of a(100) from right to left
S := convert(a(100), string):
with(ListTools):
the_List := [seq(parse(S[i]), i = 1..length(S))]:
Reverse(the_List); # Peter Bala, Nov 04 2022
  • Mathematica
    b = {6}; g[n_] := Block[{k = 0, c}, While[c = FromDigits[Prepend[b, k]]; Mod[c^2, 10^n] != c, k++ ]; b = Prepend[b, k]]; Do[ g[n], {n, 2, 105}]; Reverse[b]
With[{n = 150}, Reverse[IntegerDigits[PowerMod[16, 5^n, 10^n]]]] (* IWABUCHI Yu(u)ki, Feb 16 2024 *)
  • PARI
    {a(n)=local(b=6,v=[]);for(k=1,n+1,b=b^5%10^k;v=concat(v,(10*b\10^k)));v[n+1]} \\ Paul D. Hanna, Jul 06 2006
  • PARI
    Vecrev(digits(lift(chinese(Mod(0, 2^100), Mod(1, 5^100))))) \\ Seiichi Manyama, Aug 07 2019

Formula

x = r^4 where r=...1441224165530407839804103263499879186432 (A120817). x = 10-adic limit_{n->oo} 6^(5^n). - Paul D. Hanna, Jul 06 2006
For n >= 2, the final n+1 digits of either 2^(10^n), 4^(10^n) or 6^(10^n), when read from right to left, give the first n+1 entries in the sequence. - Peter Bala, Nov 05 2022

Extensions

More terms from David W. Wilson
Edited by David W. Wilson, Sep 26 2002

A055620 Digits of an idempotent 6-adic number.

Original entry on oeis.org

4, 4, 3, 5, 0, 2, 4, 3, 3, 3, 0, 4, 0, 0, 4, 1, 4, 2, 4, 3, 0, 0, 0, 5, 0, 3, 0, 0, 0, 2, 4, 1, 2, 2, 5, 1, 3, 3, 1, 5, 4, 2, 2, 4, 1, 5, 3, 5, 4, 3, 0, 3, 1, 5, 3, 2, 2, 5, 2, 1, 0, 0, 3, 0, 0, 1, 2, 3, 2, 4, 0, 1, 0, 1, 5, 4, 4, 5, 1, 3, 5, 4, 2, 5, 4, 0, 5, 1, 2, 0, 5, 4, 5, 3, 1, 5, 2, 1, 3, 3, 2, 3, 3, 5, 3
Offset: 0

Views

Author

Paolo Dominici (pl.dm(AT)libero.it), Jun 04 2000

Keywords

Comments

( a(0) + a(1)*6 + a(2)*6^2 + ... )^k = a(0) + a(1)*6 + a(2)*6^2 + ... for each k. Apart from 0 and 1 in base 6 there are only 2 numbers with this property. For the other see A054869.

Examples

			(a(0) + a(1)*6 + a(2)*6^2 + a(3)*6^3)^2 == (a(0) + a(1)*6 + a(2)*6^2 + a(3)*6^3) mod 6^4 because 1478656 == 1216 (mod 1296).

References

  • V. deGuerre and R. A. Fairbairn, Automorphic numbers, J. Rec. Math., 1 (No. 3, 1968), 173-179.

Links

Crossrefs

The six examples given by deGuerre and Fairbairn are A055620, A054869, A018247, A018248, A259468, A259469.

Programs

  • PARI
    first(p)=Vecrev(digits(lift(Mod(4,6^p)^3^p), 6)) \\ Charles R Greathouse IV, Nov 01 2022
  • Python
    n=10000;res=pow((3**n+1)//2,n,3**n)*2**n
for i in range(n):print(i,res%6);res//=6
# Kenny Lau, Jun 09 2018

Formula

If A is the 6-adic number, A == 4^(3^n) mod 6^n. - Robert Dawson, Oct 28 2022

A259468 Digits of an idempotent 12-adic number.

Original entry on oeis.org

4, 5, 8, 3, 11, 1, 6, 11, 1, 2, 11, 2, 5, 4, 3, 11, 7, 6, 2, 11, 6, 8, 9, 6, 1, 10, 3, 10, 1, 8, 10, 3, 3, 1, 10, 8, 6, 4, 6, 11, 2, 0, 4, 0, 3, 8, 9, 9, 5, 3, 1, 0, 9, 7, 11, 8, 7, 6, 5, 2, 5, 1, 10, 2, 11, 0, 2, 7, 6, 6, 5, 4, 7, 2, 0, 5, 11, 7, 8, 7, 2, 6
Offset: 0

Views

Author

N. J. A. Sloane, Jul 02 2015

Keywords

References

  • V. deGuerre and R. A. Fairbairn, Automorphic numbers, J. Rec. Math., 1 (No. 3, 1968), 173-179.

Links

Crossrefs

The six examples given by deGuerre and Fairbairn are A055620, A054869, A018247, A018248, A259468, A259469.

Programs

  • Sage
    def a_list(n) : return crt(1, 0, 3^n, 4^n).digits(12) # Eric M. Schmidt, Jul 09 2015

Extensions

More terms from Eric M. Schmidt, Jul 09 2015

A054869 Digits of an idempotent 6-adic number.

Original entry on oeis.org

3, 1, 2, 0, 5, 3, 1, 2, 2, 2, 5, 1, 5, 5, 1, 4, 1, 3, 1, 2, 5, 5, 5, 0, 5, 2, 5, 5, 5, 3, 1, 4, 3, 3, 0, 4, 2, 2, 4, 0, 1, 3, 3, 1, 4, 0, 2, 0, 1, 2, 5, 2, 4, 0, 2, 3, 3, 0, 3, 4, 5, 5, 2, 5, 5, 4, 3, 2, 3, 1, 5, 4, 5, 4, 0, 1, 1, 0, 4, 2, 0, 1, 3, 0, 1, 5, 0, 4, 3, 5, 0, 1, 0, 2, 4, 0, 3, 4, 2
Offset: 0

Views

Author

Paolo Dominici (pl.dm(AT)libero.it), May 23 2000

Keywords

Comments

( a(0) + a(1)*6 + a(2)*6^2 + ... )^k = a(0) + a(1)*6 + a(2)*6^2 + ... for each k. Apart from 0 and 1, in base 6 there are only 2 numbers with this property. For the other see A055620.

References

  • V. deGuerre and R. A. Fairbairn, Automorphic numbers, J. Rec. Math., 1 (No. 3, 1968), 173-179.

Links

Crossrefs

The six examples given by deGuerre and Fairbairn are A055620, A054869, A018247, A018248, A259468, A259469.

Programs

  • Python
    n=10000;res=1-pow((3**n+1)//2,n,3**n)*2**n
for i in range(n):print(i,res%6);res//=6
# Kenny Lau, Jun 09 2018

Formula

a(n) == 3^(2^n) (mod 6^n). - Robert Dawson, Oct 28 2022

A308249 Squares of automorphic numbers in base 12 (cf. A201918).

Original entry on oeis.org

0, 1, 16, 81, 4096, 6561, 263169, 1478656, 40960000, 205549569, 54988374016, 233605955584, 6263292059649, 303894740860929, 338531738189824, 170196776412774400, 709858175909625856, 18638643564726714369, 124592287100855910400, 2576097707358918017025, 479214351668445504864256
Offset: 1

Views

Author

Jeremias M. Gomes, May 17 2019

Keywords

Comments

(sqrt(m))12 is a suffix of m_12. - _A.H.M. Smeets, Aug 09 2019
All terms k^2 in this sequence (except the trivials 0 and 1) have a square root k that is the suffix of one of the 12-adic numbers given by A259468 or A259469. From this, the sequence has an infinite number of terms. - A.H.M. Smeets, Aug 09 2019

Examples

			4096 = 2454_12 and sqrt(2454_12) = 54_12. Hence 4096 is in the sequence.

Links

  • V. deGuerre and R. A. Fairbairn, Automorphic numbers, J. Rec. Math., 1 (No. 3, 1968), 173-179.

Crossrefs

Cf. A201918, A259468, A259469.

Programs

  • Python
    dig = "0123456789AB"
def To12(n):
    s = ""
    while n > 0:
        s, n = dig[n%12]+s, n//12
    return s
n, m = 1, 0
print(n,m*m)
while n < 100:
    m = m+1
    m2, m1 = To12(m*m), To12(m)
    i, i2, i1 = 0, len(m2), len(m1)
    while i < i1 and (m2[i2-i-1] == m1[i1-i-1]):
        i = i+1
    if i == i1:
        print(n,m*m)
n = n+1 # A.H.M. Smeets, Aug 09 2019
  • Sage
    [(n * n) for n in (0..1000000) if (n * n).str(base = 12).endswith(n.str(base = 12))]

Formula

Equals A201918(n)^2.

Extensions

Terms a(16)..a(21) from A.H.M. Smeets, Aug 09 2019
Showing 1-6 of 6 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.