A020330 Numbers whose base-2 representation is the juxtaposition of two identical strings.
3, 10, 15, 36, 45, 54, 63, 136, 153, 170, 187, 204, 221, 238, 255, 528, 561, 594, 627, 660, 693, 726, 759, 792, 825, 858, 891, 924, 957, 990, 1023, 2080, 2145, 2210, 2275, 2340, 2405, 2470, 2535, 2600, 2665, 2730, 2795, 2860, 2925, 2990, 3055, 3120, 3185, 3250
Offset: 1
Examples
36 is a term because 36 = 100100_2, which is 100 followed by 100.
Links
- Reinhard Zumkeller, Table of n, a(n) for n = 1..8191
- Daniel M. Kane, Carlo Sanna, and Jeffrey Shallit, Waring's Theorem for Binary Powers, Combinatorica, Vol. 39, No. 6 (2019), pp. 1335-1350, arXiv preprint, arXiv:1801.04483 [math.NT], 2018.
- Parthasarathy Madhusudan, Dirk Nowotka, Aayush Rajasekaran, and Jeffrey Shallit, Lagrange's Theorem for Binary Squares, arXiv:1710.04247 [math.NT], 2017-2018.
- Manfred Madritsch and Stephan Wagner, A central limit theorem for integer partitions, Monatshefte für Mathematik, Vol. 161, No. 1 (2010), pp. 85-114, alternative link.
- Aayush Rajasekaran, Using Automata Theory to Solve Problems in Additive Number Theory, MS thesis, University of Waterloo, 2018.
Crossrefs
Programs
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Haskell
a020330 n = foldr (\d v -> 2 * v + d) 0 (bs ++ bs) where bs = a030308_row n -- Reinhard Zumkeller, Feb 19 2013
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Magma
[n+2*n*2^Floor(Log(2, n)): n in [1..50]]; // Vincenzo Librandi, Apr 05 2018
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Maple
a:= n-> (l-> Bits[Join]([l[],l[]]))(Bits[Split](n)): seq(a(n), n=1..50); # Alois P. Heinz, Aug 24 2024
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Mathematica
Table[n + 2 n 2^Floor[Log[2, n]], {n, 50}] (* T. D. Noe, Dec 10 2013 *) FromDigits[#, 2] & /@ (# <> # & /@ IntegerString[Range@100, 2]) (* Hans Rudolf Widmer, Aug 24 2024 *) -
PARI
a(n)=n+n<<#binary(n) \\ Charles R Greathouse IV, Mar 29 2013
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PARI
is(n)=my(L=#binary(n)\2); n>>L==bitand(n,2^L-1) \\ Charles R Greathouse IV, Mar 29 2013
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Python
def a(n): return int(bin(n)[2:]*2, 2) print([a(n) for n in range(1, 51)]) # Michael S. Branicky, Mar 10 2021
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Python
def A020330(n): return (n<
Formula
a(n) = n + 2*n*2^floor(log_2(n)). - Ralf Stephan, Dec 07 2004
Sum_{n>=1} 1/a(n) = A330157. - Amiram Eldar, Oct 22 2020
a(n) = n * (2^A070939(n) + 1). - Jianing Song, Apr 10 2021
Comments