The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A003226 Automorphic numbers: m^2 ends with m. (Formerly M3752) 46
 0, 1, 5, 6, 25, 76, 376, 625, 9376, 90625, 109376, 890625, 2890625, 7109376, 12890625, 87109376, 212890625, 787109376, 1787109376, 8212890625, 18212890625, 81787109376, 918212890625, 9918212890625, 40081787109376, 59918212890625, 259918212890625, 740081787109376 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,3 COMMENTS Also called curious numbers. For entries after the second, two successive terms sum up to a total having the form 10^n + 1. - Lekraj Beedassy, Apr 29 2005 [This comment is clearly wrong as stated. The sums of two consecutive terms are 1, 6, 11, 31, 101, 452, 1001, 10001, 100001, 200001, 1000001, 3781250, .... - T. D. Noe, Nov 14 2010] If a d-digit number n is in the sequence, then so is 10^d+1-n. However, the same number can be 10^d+1-n for different n in the sequence (e.g., 10^3+1-376 = 10^4+1-9376 = 625), which spoils Beedassy's comment. - Robert Israel, Jun 19 2015 Substring of both its square and its cube not congruent to 0 (mod 10). See A029943. - Robert G. Wilson v, Jul 16 2005 a(n)^k ends with a(n) for k > 0; see also A029943. - Reinhard Zumkeller, Nov 26 2011 Apart from initial term, a subsequence of A046831. - M. F. Hasler, Dec 05 2012 This is also the sequence of numbers such that the n-th m-gonal number ends in n for any m==0,4,8,16 (mod 20). - Robert Dawson, Jul 09 2018 Apart from 6, a subsequence of A301912. - Robert Dawson, Aug 01 2018 REFERENCES J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 76, p. 26, Ellipses, Paris 2008. V. deGuerre and R. A. Fairbairn, Automorphic numbers, J. Rec. Math., 1 (No. 3, 1968), 173-179. R. A. Fairbairn, More on automorphic numbers, J. Rec. Math., 2 (No. 3, 1969), 170-174. Jan Gullberg, Mathematics, From the Birth of Numbers, W. W. Norton & Co., NY, page 253-4. B. A. Naik, 'Automorphic numbers' in 'Science Today'(subsequently renamed '2001') May 1982 pp. 59, Times of India, Mumbai. Ya. I. Perelman, Algebra can be fun, pp. 97-98. Clifford A. Pickover, A Passion for Mathematics, John Wiley & Sons, Hoboken, 2005, p. 64. C. P. Schut, Idempotents. Report AM-R9101, Centrum voor Wiskunde en Informatica, Amsterdam, 1991. N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence). LINKS Vincenzo Librandi, Table of n, a(n) for n = 1..200 Robert Dawson, On Some Sequences Related to Sums of Powers, J. Int. Seq., Vol. 21 (2018), Article 18.7.6. R. A. Fairbairn, More on automorphic numbers, J. Rec. Math., 2.3 (1969), 170-174. (Annotated scanned copy) V. deGuerre & R. A. Fairbairn, Automorphic numbers, J. Rec. Math., 1.3 (1968), 173-179 Beeler, M., Gosper, R. W. and Schroeppel, R., HAKMEM. MIT AI Memo 239, Feb 29 1972 (Item 95 provides a 40-digit member of this sequence). W. Schneider, Automorphic Numbers C. P. Schut, Idempotents, Report AM-R9101, Centre for Mathematics and Computer Science, Amsterdam, 1991. (Annotated scanned copy) Eric Weisstein's World of Mathematics, Automorphic Number Wikipedia, Automorphic number Xiaolong Ron Yu, Curious Numbers, Pi Mu Epsilon Journal, Spring 1999, pp. 819-823. FORMULA Equals {0, 1} union A007185 union A016090. MAPLE V:= proc(m) option remember;   select(t -> t^2 - t mod 10^m = 0, map(s -> seq(10^(m-1)*j+s, j=0..9), V(m-1))) end proc: V(0):= {0, 1}: V(1):= {5, 6}: sort(map(op, [V(0), seq(V(i) minus V(i-1), i=1..50)])); # Robert Israel, Jun 19 2015 MATHEMATICA f[k_] := (r = Reduce[0 < 10^k < n < 10^(k + 1) && n^2 == m*10^(k + 1) + n, {n, m}, Integers]; If[Head[r] === And, n /. ToRules[r], n /. {ToRules[r]}]); Flatten[ Join[{0, 1}, Table[f[k], {k, 0, 13}]]] (* Jean-François Alcover, Dec 01 2011 *) Union@ Join[{1}, Array[PowerMod[5, 2^#, 10^#] &, 16, 0], Array[PowerMod[16, 5^#, 10^#] &, 16, 0]] (* Robert G. Wilson v, Jul 23 2018 *) PROG (Haskell) import Data.List (isSuffixOf) a003226 n = a003226_list !! (n-1) a003226_list = filter (\x -> show x `isSuffixOf` show (x^2)) a008851_list -- Reinhard Zumkeller, Jul 27 2011 (PARI) is_A003226(n)={n<2 || 10^valuation(n^2-n, 10)>n} \\ M. F. Hasler, Dec 05 2012 (PARI) A003226(n)={ n<3 & return(n-1); my(i=10, j=10, b=5, c=6, a=b); for( k=4, n, while(b<=a, b=b^2%i*=10); while(c<=a, c=(2-c)*c%j*=10); a=min(b, c)); a } \\ M. F. Hasler, Dec 06 2012 (Sage) def automorphic(maxdigits, pow, base=10) :     morphs = [[0]]     for i in range(maxdigits):         T=[d*base^i+x for x in morphs[-1] for d in range(base)]         morphs.append([x for x in T if x^pow % base^(i+1) == x])     res = list(set(sum(morphs, []))); res.sort()     return res # call with pow=2 for this sequence, Eric M. Schmidt, Feb 09 2014 (MAGMA) [n: n in [0..10^7] | Intseq(n^2)[1..#Intseq(n)] eq Intseq(n)]; // Vincenzo Librandi, Jul 03 2015 CROSSREFS Cf. A008851, A018247, A018248, A018834, A033819, A035383, A046831, A052228. Sequence in context: A136888 A038248 A046831 * A088834 A137081 A137079 Adjacent sequences:  A003223 A003224 A003225 * A003227 A003228 A003229 KEYWORD nonn,base,nice,easy AUTHOR EXTENSIONS More terms from Michel ten Voorde, Apr 11 2001 Edited by David W. Wilson, Sep 26 2002 Incorrect statement removed from title by Robert Dawson, Jul 09 2018 STATUS approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified January 28 03:43 EST 2021. Contains 340490 sequences. (Running on oeis4.)