A135627 -id:A135627 - cp's OEIS frontend

A138148 Cyclops numbers with binary digits only.

0, 101, 11011, 1110111, 111101111, 11111011111, 1111110111111, 111111101111111, 11111111011111111, 1111111110111111111, 111111111101111111111, 11111111111011111111111, 1111111111110111111111111, 111111111111101111111111111, 11111111111111011111111111111

Offset: 0

Omar E. Pol, Mar 18 2008

All members are palindromes A002113. The first five members are mentioned in A129868.
Also, binary representation of A129868.
a(A090748(n)) is equal to A138831(n), the n-th perfect number minus 1, written in base 2.
Except for the first term (replace 0 with 1) the binary representation of the n-th iteration of the elementary cellular automaton, Rule 219 starting with a single ON (black) cell. - Robert Price, Feb 21 2016
a(1) = 101 is only prime number in this sequence since a(n) = (10^(n+1)+1)*(10^n-1)/9. - Altug Alkan, May 11 2016

			n ........ a(n) .... A129868(n): value of a(n) read in base 2.
0 ......... 0 ......... 0
1 ........ 101 ........ 5
2 ....... 11011 ....... 27
3 ...... 1110111 ...... 119
4 ..... 111101111 ..... 495
5 .... 11111011111 .... 2015
6 ... 1111110111111 ... 8127

S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 55.

Gennady Eremin, Partitioning the set of natural numbers into Mersenne trees and into arithmetic progressions; Natural Matrix and Linnik's constant, arXiv:2405.16143 [math.CO], 2024. See pp. 9, 14.
Brady Haran and Simon Pampena, Glitch Primes and Cyclops Numbers, Numberphile video, video (2015).
Eric Weisstein's World of Mathematics, Elementary Cellular Automaton
S. Wolfram, A New Kind of Science
Index to Elementary Cellular Automata
Index entries for sequences related to cellular automata
Index entries for linear recurrences with constant coefficients, signature (111,-1110,1000).

Cyclops numbers: A134808. Cf. A002113, A129868.
Cf. A000396, A090748, A135627, A138831.
Cf. A002275 (repunits R_n = (10^n-1)/9), A011557 (10^n).

[(-1 - 9*10^n + 10^(1 + 2*n))/9 : n in [0..15]]; // Wesley Ivan Hurt, Dec 08 2015

A138148:=n->(-1-9*10^n+10^(1+2*n))/9: seq(A138148(n), n=0..15); # Wesley Ivan Hurt, Dec 08 2015

Table[(-1 - 9*10^n + 10^(1 + 2*n))/9, {n, 0, 15}] (* Wesley Ivan Hurt, Dec 08 2015 *)

for(n=1, 20, if(n%2==1, c=((10^n-1)/9)-10^((n-1)/2); print1(c, ", "))) \\ Felix Fröhlich, Jul 07 2014

apply( {A138148(n)=10^(n*2+1)\9-10^n}, [0..15]) \\ M. F. Hasler, Feb 08 2020

From Colin Barker, Feb 21 2013: (Start)
a(n) = (-1-9*10^n+10^(1+2*n))/9.
G.f.: x*(200*x-101) / ((x-1)*(10*x-1)*(100*x-1)). (End)
a(n) = 111*a(n-1) - 1110*a(n-2) + 1000*a(n-3) for n>2. - Wesley Ivan Hurt, Dec 08 2015
a(n) = A000533(n+1)*A002275(n). - Altug Alkan, May 12 2016
E.g.f.: (-1 - 9*exp(9*x) + 10*exp(99*x))*exp(x)/9. - Ilya Gutkovskiy, May 12 2016
a(n) = A002275(2n+1) - A011557(n). - M. F. Hasler, Feb 08 2020

More terms from Omar E. Pol, Feb 09 2020

A002503 Numbers k such that binomial(2*k,k) is divisible by (k+1)^2.

Original entry on oeis.org

5, 14, 27, 41, 44, 65, 76, 90, 109, 125, 139, 152, 155, 169, 186, 189, 203, 208, 209, 219, 227, 230, 237, 265, 275, 298, 307, 311, 314, 321, 324, 329, 344, 377, 413, 419, 428, 434, 439, 441, 449, 458, 459, 467, 475

Offset: 1

N. J. A. Sloane, Mira Bernstein

From Amiram Eldar, Mar 28 2021: (Start)
Balakram (1929) proved that:
1) This sequence is infinite.
2) If m is an even perfect number (A000396) then m-1 is a term.
3) If m = p*q - 1, where p and q are primes, and (3/2)*p < q < 2*p, then m is a term.
4) m is a term if and only if Sum_{k>=1} floor(2*m/p^k) >= 2 * Sum_{k>=1} floor((m+1)/p^k), for all primes p. (End)

Hoon Balakram, On the values of n which make (2n)!/(n+1)!(n+1)! an integer, J. Indian Math. Soc., Vol. 18 (1929), pp. 97-100.
Thomas Koshy, Catalan numbers with applications, Oxford University Press, 2008, pp. 69-70.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from T. D. Noe)
P. Erdős, R. L. Graham, I. Z. Russa and E. G. Straus, On the prime factors of C(2n,n), Math. Comp., Vol. 29, No. 129 (1975), pp. 83-92.
Carl Pomerance, Divisors of the middle binomial coefficient, Amer. Math. Monthly, Vol. 112, No. 7 (2015), pp. 636-644; alternative link.

Positions of zeros in A065350.
Cf. A000108, A065344-A065349.
Equals A067348(n+2)/2 - 1.
Cf. A000396, A135627.

import Data.List (elemIndices)
a002503 n = a002503_list !! (n-1)
a002503_list = map (+ 1) $ elemIndices 0 a065350_list
-- Reinhard Zumkeller, Sep 16 2014

Select[Range[500],Divisible[Binomial[2#,#],(#+1)^2]&] (* Harvey P. Dale, May 21 2012 *)

isok(n) = binomial(2*n, n) % (n+1)^2 == 0; \\ Michel Marcus, Jan 11 2016

A065350(a(n)) = 0. - Reinhard Zumkeller, Sep 16 2014

Balakram reference corrected by T. D. Noe, Jan 16 2007

A138831 n-th perfect number minus 1, written in base 2.

Original entry on oeis.org

101, 11011, 111101111, 1111110111111, 1111111111110111111111111, 111111111111111101111111111111111, 1111111111111111110111111111111111111

Offset: 1

Omar E. Pol, Apr 08 2008, Apr 09 2008, Apr 14 2008

Subset of A138148, cyclops numbers with binary digits, only.
Subset of A002113, palindromes in base 10.
a(n) has 2*A090748(n) digits 1.
The number of digits of a(n) is 2*A000043(n)-1, equal to A133033(n), the number of proper divisors of n-th perfect number.
a(n) = (A135627(n) written in base 2).

			n ... A000396(n) - 1 = A135627(n) ............. a(n)
1 ............ 6 - 1 = ...... 5 ............... 101
2 ........... 28 - 1 = ..... 27 .............. 11011
3 .......... 496 - 1 = .... 495 ............ 111101111
4 ......... 8128 - 1 = ... 8127 .......... 1111110111111
5 ..... 33550336 - 1 = 33550335 .... 1111111111110111111111111

Omar E. Pol, Determinacion geometrica de los numeros primos y perfectos.

Cf. A000043, A000396, A090748, A133033, A134808, A135627, A138131, A138148.

a(n) = A138148(A090748(n)).

A135629 Perfect numbers plus 1.

Original entry on oeis.org

7, 29, 497, 8129, 33550337, 8589869057, 137438691329, 2305843008139952129, 2658455991569831744654692615953842177, 191561942608236107294793378084303638130997321548169217

Offset: 1

Omar E. Pol, Nov 27 2007

G. C. Greubel, Table of n, a(n) for n = 1..15

Cf. A000396, A135627.

(# (# + 1))/2 & /@ Select[2^Range[100] - 1, PrimeQ] + 1 (* G. C. Greubel, Oct 24 2016, based on Harvey P. Dale code from A000396 *)

a(n) = A000396(n) + 1.

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A138148 Cyclops numbers with binary digits only.

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A002503 Numbers k such that binomial(2*k,k) is divisible by (k+1)^2.

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A138831 n-th perfect number minus 1, written in base 2.

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A135629 Perfect numbers plus 1.

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