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