A005367 -id:A005367 - cp's OEIS frontend

A092431 Numbers having in binary representation a leading 1 followed by n zeros and n-1 ones.

2, 9, 35, 135, 527, 2079, 8255, 32895, 131327, 524799, 2098175, 8390655, 33558527, 134225919, 536887295, 2147516415, 8590000127, 34359869439, 137439215615, 549756338175, 2199024304127, 8796095119359, 35184376283135, 140737496743935, 562949970198527

Offset: 1

Reinhard Zumkeller, Mar 23 2004

Smallest numbers having in binary representation n 0's and n 1's: a(n) = Min{m: A023416(m)=A000120(m)=n}.

Vincenzo Librandi, Table of n, a(n) for n = 1..500
Eric Weisstein's World of Mathematics, Binary
Index entries for linear recurrences with constant coefficients, signature (7,-14,8).

Cf. A007088, A001700.
Subsequence of A031443.
Cf. A000120, A023416.
Cf. A007582, A056326, A005367, A063376, A032125, A056309, A028403, A001576, A028400, A049775.

LinearRecurrence[{7, -14, 8}, {2, 9, 35}, 40] (* Vladimir Joseph Stephan Orlovsky, Feb 20 2012 *)
Table[FromDigits[Join[PadRight[{1},n,0],PadRight[{},n-2,1]],2],{n,2,30}]//Sort (* or *) Rest[CoefficientList[Series[x (-2+5x)/((x-1)(2x-1)(4x-1)),{x,0,30}],x]] (* Harvey P. Dale, Jul 30 2021 *)

a(n+1) = 2*a(n) + 4^n + 1.
a(n) = 2^(2*n-1) + 2^(n-1) - 1.
a(n) = A007582(n)-1 = A056326(2n+1) = A005367(n-1)/2 = A063376(n)/2-1 = A032125(n+1)/3-1 = A056309(2n+1)/2 = A028403(n+1)/4-1 = (A001576(n)-3)/2 = (A028400(n+1)-9)/8 = Sum_{k=2..n+1} A049775(k). - Ralf Stephan, Mar 24 2004
G.f.: x*(-2+5*x) / ( (x-1)*(2*x-1)*(4*x-1) ). - R. J. Mathar, Jun 01 2011
E.g.f.: exp(x)*(exp(3*x) + exp(x) - 2)/2. - Stefano Spezia, Sep 27 2023

A053144 Cototient of the n-th primorial number.

Original entry on oeis.org

1, 4, 22, 162, 1830, 24270, 418350, 8040810, 186597510, 5447823150, 169904387730, 6317118448410, 260105476071210, 11228680258518030, 529602053223499410, 28154196550210460730, 1665532558389396767070

Offset: 1

Labos Elemer, Feb 28 2000

nonn

a(n) > A005367(n), a(n) > A002110(n)/2.

Limit_{n->oo} a(n)/A002110(n) = 1 because (in the limit) the quotient is the probability that a randomly selected integer contains at least one of the first n primes in its factorization. - Geoffrey Critzer, Apr 08 2010

			In the reduced residue system of q(4) = 2*3*5*7 - 210 the number of coprimes to 210 is 48, while a(4) = 210 - 48 = 162 is the number of values divisible by one of the prime factors of q(4).

Michael De Vlieger, Table of n, a(n) for n = 1..350

Cf. A002110, A051953, A005867, A038110, A038111, A174909, A293558.
Cf. A000040 (prime numbers).
Cf. A161527, A060753.
Column 1 of A281891.

Abs[Table[ Total[Table[(-1)^(k + 1)* Total[Apply[Times, Subsets[Table[Prime[n], {n, 1, m}], {k}], 2]], {k, 0, m - 1}]], {m, 1, 22}]] (* Geoffrey Critzer, Apr 08 2010 *)
Array[# - EulerPhi@ # &@ Product[Prime@ i, {i, #}] &, 17] (* Michael De Vlieger, Feb 17 2019 *)

a(n) = prod(k=1, n, prime(k)) - prod(k=1, n, prime(k)-1); \\ Michel Marcus, Feb 08 2019

a(n) = A051953(A002110(n)) = A002110(n) - A005867(n).
a(n) = a(n-1)*A000040(n) + A005867(n-1). - Bob Selcoe, Feb 21 2016
a(n) = (1/A000040(n+1) - A038110(n+1)/A038111(n+1))*A002110(n+1). - Jamie Morken, Feb 08 2019
a(n) = A161527(n)*A002110(n)/A060753(n+1). - Jamie Morken, May 13 2022

cp's OEIS Frontend

A092431 Numbers having in binary representation a leading 1 followed by n zeros and n-1 ones.

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