A081252 -id:A081252 - cp's OEIS frontend

A081254 Numbers k such that A081252(m)/m^2 has a local maximum for m = k.

1, 3, 6, 13, 26, 53, 106, 213, 426, 853, 1706, 3413, 6826, 13653, 27306, 54613, 109226, 218453, 436906, 873813, 1747626, 3495253, 6990506, 13981013, 27962026, 55924053, 111848106, 223696213, 447392426, 894784853, 1789569706, 3579139413

Offset: 1

Klaus Brockhaus, Mar 17 2003

The limit of the local maxima, lim_{m->inf} A081252(m)/m^2 = 1/10. For local minima cf. A081253.

Row sums of the triangle A181971. - Reinhard Zumkeller, Jul 09 2012

			13 is a term since A081252(12)/12^2 = 15/144 = 0.104..., A081252(13)/13^2 = 18/169 = 0.106..., A081252(14)/14^2 = 20/196 = 0.102....

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Thomas Baruchel, Properties of the cumulated deficient binary digit sum, arXiv:1908.02250 [math.NT], 2019.
Klaus Brockhaus, Illustration for A053646, A081252, A081253 and A081254
Index entries for linear recurrences with constant coefficients, signature (2,1,-2).

Cf. A000975, A020714, A020989, A048573, A052549, A053646, A072197, A081252, A081253, A266219 (binary).

[Floor(2^(n-1)*5/3): n in [1..40]]; // Vincenzo Librandi, Apr 04 2012

seq(floor(2^(n-1)*5/3),n=1..35); # Muniru A Asiru, Sep 20 2018

Rest@CoefficientList[Series[-(x^2 - x - 1)*x/((x - 1)*(x + 1)*(2*x - 1)), {x, 0, 32}], x] (* Vincenzo Librandi, Apr 04 2012 *)
a[n_]:=Floor[2^(n-1)*5/3]; Array[a,33,1] (* Stefano Spezia, Sep 01 2018 *)

a(n) = 2^(n-1)*5\3; \\ Altug Alkan, Sep 21 2018

a(n) = floor(2^(n-1)*5/3). [corrected by Michel Marcus, Sep 21 2018]
a(n) = a(n-2) + 5*2^(n-3) for n > 2;
a(n+2) - a(n) = A020714(n-1);
a(n) + a(n-1) = A052549(n-1) for n > 1;
a(2*n+1) = A020989(n); a(2n) = A072197(n-1);
a(n+1) - a(n) = A048573(n-1).
G.f.: -(x^2 - x - 1)*x/((x - 1)*(x + 1)*(2*x - 1)).
a(n) = 5*2^(n-1)/3 + (-1)^n/6-1/2. a(n) = 2*a(n-1) + (1+(-1)^n)/2, a(1)=1. - Paul Barry, Mar 24 2003
a(2n) = 2*a(2*n-1) + 1, a(2*n+1) = 2*a(2*n), a(1)=1. a(n) = A000975(n-1) + 2^(n-1). - Philippe Deléham, Oct 15 2006
a(n) = A005578(n) + A000225(n-1). - Yuchun Ji, Sep 21 2018
a(n) - a(n-2) = 2 * (a(n-1) - a(n-3)), with a(0..2)=[1,3,6]. - Yuchun Ji, Mar 18 2020

A081253 Numbers k such that A081252(m)/m^2 has a local minimum for m = k.

Original entry on oeis.org

2, 4, 9, 18, 37, 74, 149, 298, 597, 1194, 2389, 4778, 9557, 19114, 38229, 76458, 152917, 305834, 611669, 1223338, 2446677, 4893354, 9786709, 19573418, 39146837, 78293674, 156587349, 313174698, 626349397, 1252698794, 2505397589

Offset: 1

Klaus Brockhaus, Mar 17 2003

The limit of the local minima, lim_{n->infinity} A081252(n)/n^2 = 1/14. For local maxima cf. A081254.

			9 is a term since A081252(8)/8^2 = 5/64 = 0.078, A081252(9)/9^2 = 6/81 = 0.074, A081252(10)/10^2 = 8/100 = 0.080.

Michael De Vlieger, Table of n, a(n) for n = 1..3321
Klaus Brockhaus, Illustration for A053646, A081252, A081253 and A081254
Chris J. Mitchell and Peter R. Wild, Constructing orientable sequences, arXiv:2108.03069 [math.CO], 2021. See Table 2 p. 12 but with different offset.
Index entries for linear recurrences with constant coefficients, signature (2,1,-2).

Cf. A053646, A081252, A081254, A005009, A062092.

Cf. A266071 (binary).

Rest@ CoefficientList[Series[-x (x^2 - 2)/((x - 1) (x + 1) (2 x - 1)), {x, 0, 31}], x]

print([7*2**n//6 for n in range(1, 50)]) # Karl V. Keller, Jr., May 22 2022

a(n) = floor(2^(n-1)*7/3).
a(n) = a(n-2) + 7*2^(n-3) for n > 2; a(n+2) - a(n) = A005009(n-1); a(n+1) - a(n) = A062092(n-1).
G.f.: -x*(x^2 - 2)/((x - 1)*(x + 1)*(2*x - 1)).
a(n) = 2*a(n-1) for even n, otherwise a(n) = 2*a(n-1)+1, with a(1)=2. - Bruno Berselli, Jun 19 2014

Formulas adjusted to be consistent with offset 1 by Pontus von Brömssen, Sep 27 2021

A053646 Distance to nearest power of 2.

Original entry on oeis.org

0, 0, 1, 0, 1, 2, 1, 0, 1, 2, 3, 4, 3, 2, 1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 7, 6, 5, 4, 3, 2, 1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25

Offset: 1

Henry Bottomley, Mar 22 2000

Sum_{j=1..2^(k+1)} a(j) = A002450(k) = (4^k - 1)/3. - Klaus Brockhaus, Mar 17 2003

			a(10)=2 since 8 is closest power of 2 to 10 and |8-10| = 2.

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Klaus Brockhaus, Illustration for A053646, A081252, A081253 and A081254
Hsien-Kuei Hwang, S. Janson, and T.-H. Tsai, Exact and asymptotic solutions of the recurrence f(n) = f(floor(n/2)) + f(ceiling(n/2)) + g(n): theory and applications, Preprint 2016.
Hsien-Kuei Hwang, S. Janson, and T.-H. Tsai, Exact and Asymptotic Solutions of a Divide-and-Conquer Recurrence Dividing at Half: Theory and Applications, ACM Transactions on Algorithms, 13:4 (2017), #47; DOI: 10.1145/3127585.
Index entries for sequences related to distance to nearest element of some set

Cf. A006165, A053188, A060973, A081134, A002450, A081252, A081253, A081254.

a:= n-> (h-> min(n-h, 2*h-n))(2^ilog2(n)):
seq(a(n), n=1..100);  # Alois P. Heinz, Mar 28 2021

np2[n_]:=Module[{min=Floor[Log[2,n]],max},max=min+1;If[2^max-nHarvey P. Dale, Feb 21 2012 *)

a(n)=vecmin(vector(n,i,abs(n-2^(i-1))))

for(n=1,89,p=2^floor(0.1^25+log(n)/log(2)); print1(min(n-p,2*p-n),","))

a(n) = my (p=#binary(n)); return (min(n-2^(p-1), 2^p-n)) \\ Rémy Sigrist, Mar 24 2018

def A053646(n): return min(n-(m := 2**(len(bin(n))-3)),2*m-n) # Chai Wah Wu, Mar 08 2022

a(2^k+i) = i for 1 <= i <= 2^(k-1); a(3*2^k+i) = 2^k-i for 1 <= i <= 2^k; (Sum_{k=1..n} a(k))/n^2 is bounded. - Benoit Cloitre, Aug 17 2002
a(n) = min(n-2^floor(log(n)/log(2)), 2*2^floor(log(n)/log(2))-n). - Klaus Brockhaus, Mar 08 2003
From Peter Bala, Aug 04 2022: (Start)
a(n) = a( 1 + floor((n-1)/2) ) + a( ceiling((n-1)/2) ).
a(2*n) = 2*a(n); a(2*n+1) = a(n) + a(n+1) for n >= 2. Cf. A006165. (End)
a(n) = 2*A006165(n) - n for n >= 2. - Peter Bala, Sep 25 2022

A179896 Sum of the numbers between k := n-th nonprime and 2k (like a jump in a Sieve of Eratosthenes).

Original entry on oeis.org

0, 18, 45, 84, 108, 135, 198, 273, 315, 360, 459, 570, 630, 693, 828, 900, 975, 1053, 1134, 1305, 1488, 1584, 1683, 1785, 1890, 2109, 2223, 2340, 2583, 2838, 2970, 3105, 3384, 3528, 3675, 3825, 3978, 4293, 4455, 4620, 4788, 4959, 5310, 5673, 5859, 6048, 6240, 6435

Offset: 1

Odimar Fabeny, Jul 31 2010

The values 4, 7, 10... (A016777 for n>1) are the values of floor( a(k)/ A018252(k) ) where k runs through the indices where A179879(k) mod A018252(k) != 0. - Odimar Fabeny.

Proof: a(k)/A018252(k) is 3*(A081252(k)-1)/2. This is a non-integer iff A018252(k) is even. Since the n-th even nonprime is 2*n+2, floor(3*(2*n+1)/2) = 3*n+1=a(n). - Robert Israel, Aug 27 2014

			0(0) = 0, 1(2) = 0, 4(8) = 5,6,7 = 18, 6(12) = 7,8,9,10,11 = 45 and so on.

Nathaniel Johnston, Table of n, a(n) for n = 1..10000

Cf. A000040, A179545.

ithnonprime := proc(n)local k: option remember: if(n=1)then return 1: else k := procname(n-1)+1: while true do if(not isprime(k))then return k fi: k:=k+1: od: fi: end:
A179896 := proc(n)local k: k:=ithnonprime(n): return 3*k*(k-1)/2: end:
seq(A179896(n),n=1..40); # Nathaniel Johnston, Apr 21 2011

f[n_] := Plus @@ Range[n + 1, 2 n - 1]; f /@ Select[ Range@ 64, ! PrimeQ@# &] (* Robert G. Wilson v, Sep 02 2010 *)

a(n) = A045943(A141468(n+1)-1). - R. J. Mathar, Sep 01 2010

More terms from Odimar Fabeny, Aug 11 2010

Offset adapted to A141468 and to match another 0 - R. J. Mathar, Sep 01 2010

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A081254 Numbers k such that A081252(m)/m^2 has a local maximum for m = k.

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A081253 Numbers k such that A081252(m)/m^2 has a local minimum for m = k.

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A053646 Distance to nearest power of 2.

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A179896 Sum of the numbers between k := n-th nonprime and 2k (like a jump in a Sieve of Eratosthenes).

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