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 A000042 Unary representation of natural numbers. (Formerly M4804) 84
 1, 11, 111, 1111, 11111, 111111, 1111111, 11111111, 111111111, 1111111111, 11111111111, 111111111111, 1111111111111, 11111111111111, 111111111111111, 1111111111111111, 11111111111111111, 111111111111111111, 1111111111111111111, 11111111111111111111 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS Or, numbers written in base 1. If p is a prime > 5 then d_{a(p)} == 1 (mod p) where d_{a(p)} is a divisor of a(p). This also gives an alternate elementary proof of the infinitude of prime numbers by the fact that for every prime p there exists at least one prime of the form k*p + 1. - Amarnath Murthy, Oct 05 2002 11 = 1*9 + 2; 111 = 12*9 + 3; 1111 = 123*9 + 4; 11111 = 1234*9 + 5; 111111 = 12345*9 + 6; 1111111 = 123456*9 + 7; 11111111 = 1234567*9 + 8; 111111111 = 12345678*9 + 9. - Vincenzo Librandi, Jul 18 2010 Also numbers with no zeros in binary representation. - Reinhard Zumkeller, Mar 08 2015 REFERENCES K. G. Kroeber, Mathematik der Palindrome; p. 348; 2003; ISBN 3 499 615762; Rowohlt Verlag; Germany. Amarnath Murthy and Charles Ashbacher, Generalized Partitions and Some New Ideas on Number Theory and Smarandache Sequences, Hexis, Phoenix; USA 2005. See Section 2.12. D. Olivastro, Ancient Puzzles. Bantam Books, NY, 1993, p. 276. N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence). LINKS David Wasserman, Table of n, a(n) for n = 1..1000 Makoto Kamada, Factorizations of 11...11 (Repunit). Amarnath Murthy, On the divisors of Smarandache Unary Sequence. Smarandache Notions Journal, Vol. 11, No. 1-2-3, Spring 2000, page 184 Index entries for linear recurrences with constant coefficients, signature (11,-10). FORMULA a(n) = (10^n - 1)/9. G.f.: 1/((1-x)*(1-10*x)). Binomial transform of A003952. - Paul Barry, Jan 29 2004 From Paul Barry, Aug 24 2004: (Start) a(n) = 10*a(n-1) + 1, n > 1, a(1)=1. [Offset 1.] a(n) = Sum_{k=0..n} binomial(n+1, k+1)*9^k. [Offset 0.] (End) a(2n) - 2*a(n) = (3*a(n))^2. E.g., a(6) - 2*a(3) = (3*a(3))^2: 111111 - 222 = 110889. - Amarnath Murthy, Jul 21 2003 a(n) is the binary representation of the n-th Mersenne number (A000225). - Ross La Haye, Sep 13 2003 The Hankel transform of this sequence is [1,-10,0,0,0,0,0,0,0,0,...]. - Philippe Deléham, Nov 21 2007 E.g.f.: (exp(10*x) - exp(x))/9. - G. C. Greubel, Nov 04 2018 MAPLE a:= n-> parse(cat(1\$n)): seq(a(n), n=1..25);  # Alois P. Heinz, Mar 23 2018 MATHEMATICA Table[(10^n - 1)/9, {n, 1, 18}] FromDigits/@Table[PadLeft[{}, n, 1], {n, 20}] (* Harvey P. Dale, Aug 21 2011 *) PROG (PARI) a(n)=if(n<0, 0, (10^n-1)/9) (Sage) [gaussian_binomial(n, 1, 10) for n in xrange(1, 19)] # Zerinvary Lajos, May 28 2009 (Haskell) A000042 n = (10^n-1) `div` 9 -- James Spahlinger, Oct 08 2012 (Common Lisp) (defun a000042 (n) (truncate (expt 10 n) 9)) ; James Spahlinger, Oct 12 2012 (MAGMA) [(10^n - 1)/9: n in [1..20]]; // G. C. Greubel, Nov 04 2018 CROSSREFS Cf. A002275, A007088, A007089, A007090, A007091, A007092, A007093, A007094, A007095. Sequence in context: A113589 A135463 * A002275 A294348 A078998 A078191 Adjacent sequences:  A000039 A000040 A000041 * A000043 A000044 A000045 KEYWORD base,easy,nonn,nice AUTHOR EXTENSIONS More terms from Paul Barry, Jan 29 2004 STATUS approved

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Last modified November 21 00:36 EST 2018. Contains 317427 sequences. (Running on oeis4.)