A055620 -id:A055620 - cp's OEIS frontend

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

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

Yoshihide Tamori (yo(AT)salk.edu)

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

			x = ...0863811000557423423230896109004106619977392256259918212890625.

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.

Eric M. Schmidt, Table of n, a(n) for n = 0..9999
Anthony Edey, Automorphic numbers, taken from Madachy's Mathematical Recreations, Dover 1979.
V. deGuerre and R. A. Fairbairn, Automorphic numbers, Jnl. Rec. Math., 1 (No. 3, 1968), 173-179.
MathOverflow, Distribution of digits of pq-adic idempotents (aka “automorphic numbers”), 2014.
Eric Weisstein's World of Mathematics, Automorphic numbers.
Index entries for sequences related to automorphic numbers

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.

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 *)

a(n)=local(t=5);for(k=1,n+1,t=t^2%10^k);t\10^n \\ Paul D. Hanna, Jul 08 2006

Vecrev(digits(lift(chinese(Mod(1, 2^100), Mod(0, 5^100))))) \\ Seiichi Manyama, Aug 07 2019

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

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

Yoshihide Tamori (yo(AT)salk.edu)

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

			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)

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.

Seiichi Manyama, Table of n, a(n) for n = 0..9999 (terms 0..999 from Paul D. Hanna).
Anonymous, Automorphic numbers (2)
Peter Bala, A note on A018248
V. deGuerre and R. A. Fairbairn, Automorphic numbers, Jnl. Rec. Math., 1 (No. 3, 1968), 173-179.
MathOverflow, Distribution of digits of pq-adic idempotents (a.k.a. "automorphic numbers"), 2014.
Eric Weisstein's World of Mathematics, Automorphic numbers (1)
Index entries for sequences related to automorphic numbers

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.

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

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 *)

{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

Vecrev(digits(lift(chinese(Mod(0, 2^100), Mod(1, 5^100))))) \\ Seiichi Manyama, Aug 07 2019

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

More terms from David W. Wilson

Edited by David W. Wilson, Sep 26 2002

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

N. J. A. Sloane, Jul 02 2015

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

Eric M. Schmidt, Table of n, a(n) for n = 0..9999
V. deGuerre and R. A. Fairbairn, Automorphic numbers, Jnl. Rec. Math., 1 No. 3, (1968), 173-179.

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

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

More terms from Eric M. Schmidt, Jul 09 2015

A259469 Digits of an idempotent 12-adic number.

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane, Jul 02 2015

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

Eric M. Schmidt, Table of n, a(n) for n = 0..9999
V. deGuerre and R. A. Fairbairn, Automorphic numbers, Jnl. Rec. Math., 1 (No. 3, 1968), 173-179.

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

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

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

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

( 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.

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

Kenny Lau, Table of n, a(n) for n = 0..9999
V. deGuerre and R. A. Fairbairn, Automorphic numbers, Jnl. Rec. Math., 1 (No. 3, 1968), 173-179

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

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

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

A283754 The smallest number k such that k*2^n mod 3^n = 1.

Original entry on oeis.org

2, 7, 17, 76, 38, 262, 1589, 4075, 11879, 35464, 17732, 363160, 181580, 90790, 9611333, 11980120, 92083502, 175181914, 862431935, 2174608168, 1087304084, 543652042, 271826021, 235493860078, 117746930039, 1329806379184, 664903189592, 332451594796, 166225797398, 68713490263582, 446139009321089

Offset: 1

Joe Slater, Mar 23 2017

nonn

a(n) is the coefficient "a" in the Diophantine equation with two coefficients a and b, a * 2^n - b * 3^n = 1.

			2 * 2^1 mod 3^1 = 1, 7 * 2^2 mod 3^2 =1, 17 * 2^3 mod 3^3 = 1...

Robert Israel, Table of n, a(n) for n = 1..2095

Cf. A055620.

seq(2^(-n) mod 3^n, n=1..100); # Robert Israel, Mar 28 2017

Table[ PowerMod[ (3^n +1)/2, n, 3^n], {n, 30}] (* Robert G. Wilson v, Mar 28 2017 *)

a(n)= my(z=3^n); lift( Mod((z + 1)/2, z)^n); \\ Joerg Arndt, Mar 24 2017

a(n) = ((3^n + 1)/2)^n mod 3^n (proved).

Conjecture: 2*a(n+1)-a(n) = 3^n * A055620(n). - Robert Israel, Mar 28 2017

Corrected and more terms from Joerg Arndt, Mar 24 2017

cp's OEIS Frontend

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

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

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A259468 Digits of an idempotent 12-adic number.

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A259469 Digits of an idempotent 12-adic number.

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A054869 Digits of an idempotent 6-adic number.

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A283754 The smallest number k such that k*2^n mod 3^n = 1.

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Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

Extensions