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

Author

Yoshihide Tamori (yo(AT)salk.edu)

Keywords

• base
• nonn

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

• 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

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