A053646 -id:A053646 - cp's OEIS frontend

A081252 Partial sums of A053646.

0, 0, 1, 1, 2, 4, 5, 5, 6, 8, 11, 15, 18, 20, 21, 21, 22, 24, 27, 31, 36, 42, 49, 57, 64, 70, 75, 79, 82, 84, 85, 85, 86, 88, 91, 95, 100, 106, 113, 121, 130, 140, 151, 163, 176, 190, 205, 221, 236, 250, 263, 275, 286, 296, 305, 313, 320, 326, 331, 335, 338, 340, 341, 341

Offset: 1

Klaus Brockhaus, Mar 17 2003

			First seven terms of A053646 are 0,0,1,0,1,2,1, so a(7) = 5.

Klaus Brockhaus, Illustration for A053646, A081252, A081253 and A081254

Cf. A053646, A081253, A081254.

{s=0; for(n=1,65,p=2^floor(0.1^25+log(n)/log(2)); print1(s=s+min(n-p,2*p-n),","))}

a(n) = sum{j=1..n, A053646(j)}.

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

Original entry on oeis.org

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

A005942 a(2n) = a(n) + a(n+1), a(2n+1) = 2a(n+1), if n >= 2.

Original entry on oeis.org

1, 2, 4, 6, 10, 12, 16, 20, 22, 24, 28, 32, 36, 40, 42, 44, 46, 48, 52, 56, 60, 64, 68, 72, 76, 80, 82, 84, 86, 88, 90, 92, 94, 96, 100, 104, 108, 112, 116, 120, 124, 128, 132, 136, 140, 144, 148, 152, 156, 160, 162, 164, 166, 168, 170, 172, 174, 176, 178, 180, 182

Offset: 0

N. J. A. Sloane, Jeffrey Shallit

a(n) is the subword complexity (or factor complexity) of Thue-Morse sequence A010060, that is, the number of factors of length n in A010060. See Allouche-Shallit (2003). - N. J. A. Sloane, Jul 10 2012

J.-P. Allouche and J. Shallit, Automatic Sequences, Cambridge Univ. Press, 2003. See Problem 10, p. 335. - From N. J. A. Sloane, Jul 10 2012
J. Berstel et al., Combinatorics on Words: Christoffel Words and Repetitions in Words, Amer. Math. Soc., 2008. See p. 83.
Michel Rigo, Formal Languages, Automata and Numeration Systems, 2 vols., Wiley, 2014. Mentions this sequence - see "List of Sequences" in Vol. 2.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 0..1000
S. Brlek, Enumeration of factors in the Thue-Morse word, Discrete Applied Math. 24 (1989), 83-96.
A. de Luca and S. Varricchio, Some combinatorial properties of the Thue-Morse sequence and a problem in semigroups, Theoret. Comput. Sci. 63 (1989), 333-348.
Jakub Konieczny and Clemens Müllner, Arithmetical subword complexity of automatic sequences, arXiv:2309.03180 [math.NT], 2023.
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.
Luís M. S. Russo, Ana D. Correia, Gonzalo Navarro, and Alexandre P. Francisco, Approximating Optimal Bidirectional Macro Schemes, arXiv:2003.02336 [cs.DS], 2020.

Cf. A005943, A006697, A010060, A275202.

import Data.List (transpose)
a005942 n = a005942_list !! n
a005942_list = 1 : 2 : 4 : 6 : zipWith (+) (drop 6 ts) (drop 5 ts) where
   ts = concat $ transpose [a005942_list, a005942_list]
-- Reinhard Zumkeller, Nov 15 2012

a[0] = 1; a[1] = 2; a[2] = 4; a[3] = 6; a[n_?EvenQ] := a[n] = a[n/2] + a[n/2 + 1]; a[n_?OddQ]  := a[n] = 2*a[(n + 1)/2]; Array[a,60,0] (* Jean-François Alcover, Apr 11 2011 *)

a(n)=if(n<4,2*max(0,n)+(n==0),if(n%2,2*a((n+1)/2),a(n/2)+a(n/2+1)))

a(n) = 2*(A006165(n-1) + n - 1), n > 1.
G.f. (1+x^2)/(1-x)^2 + 2*x^2/(1-x)^2 * Sum_{k>=0} (x^2^(k+1)-x^(3*2^k)). - Ralf Stephan, Jun 04 2003
For n > 2, a(n) = 3*(n-1) + A053646(n-1). - Max Alekseyev, May 15 2011
a(n) = 2*A275202(n-1) for n > 1. - N. J. A. Sloane, Jun 04 2019

Typo in definition corrected by Reinhard Zumkeller, Nov 15 2012

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

A081134 Distance to nearest power of 3.

Original entry on oeis.org

0, 1, 0, 1, 2, 3, 2, 1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 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, 26, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 1, 2, 3, 4, 5, 6, 7

Offset: 1

Klaus Brockhaus, Mar 08 2003

			a(7) = 2 since 9 is closest power of 3 to 7 and 9 - 7 = 2.

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Klaus Brockhaus, Illustration for A081134, A081249, A081250 and A081251
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. A002452, A003605, A006166, A053646, A062153, A081249, A081250, A081251.

a:= n-> (h-> min(n-h, 3*h-n))(3^ilog[3](n)):
seq(a(n), n=1..100);  # Alois P. Heinz, Mar 28 2021

Flatten[Table[Join[Range[0,3^n],Range[3^n-1,1,-1]],{n,0,4}]] (* Harvey P. Dale, Dec 31 2013 *)

a(n) = my (p=#digits(n,3)); return (min(n-3^(p-1), 3^p-n)) \\ Rémy Sigrist, Mar 24 2018

def A081134(n):
    kmin, kmax = 0,1
    while 3**kmax <= n:
        kmax *= 2
    while True:
        kmid = (kmax+kmin)//2
        if 3**kmid > n:
            kmax = kmid
        else:
            kmin = kmid
        if kmax-kmin <= 1:
            break
    return min(n-3**kmin, 3*3**kmin-n) # Chai Wah Wu, Mar 31 2021

a(n) = min(n-3^floor(log(n)/log(3)), 3*3^floor(log(n)/log(3))-n).
From Peter Bala, Sep 30 2022: (Start)
a(n) = n - A006166(n); a(n) = 2*n - A003605(n).
a(1) = 0, a(2) = 1, a(3) = 0; thereafter, a(3*n) = 3*a(n), a(3*n+1) = 2*a(n) + a(n+1) and a(3*n+2) = a(n) + 2*a(n+1). (End)

A060973 a(2n+1) = a(n+1)+a(n), a(2n) = 2*a(n), with a(1)=0 and a(2)=1.

Original entry on oeis.org

0, 1, 1, 2, 2, 2, 3, 4, 4, 4, 4, 4, 5, 6, 7, 8, 8, 8, 8, 8, 8, 8, 8, 8, 9, 10, 11, 12, 13, 14, 15, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32

Offset: 1

Henry Bottomley, May 09 2001

			a(6) = 2*a(3) = 2*1 = 2.
a(7) = a(3) + a(4) = 1 + 2 = 3.

R. J. Mathar, Table of n, a(n) for n = 1..1000
Max A. Alekseyev, Enumeration of Payphone Permutations, arXiv:2304.04324 [math.CO], 2023.
Michael De Vlieger, Log-log scatterplot of a(n), n = 1..2^16.
H.-K. 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.
H.-K. 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
Jeffrey Shallit, Intertwining of Complementary Thue-Morse Factors, arXiv:2203.02917 [cs.FL], 2022.
Ralf Stephan, Some divide-and-conquer sequences ....
Ralf Stephan, Table of generating functions.

Cf. A006165, A053646.

A060973 := proc(n)
    option remember;
    if n <= 2 then
        return n-1;
    fi;
    if n mod 2 = 0 then
        2*procname(n/2)
    else
        procname((n-1)/2)+procname((n+1)/2);
    fi;
end proc:  # R. J. Mathar Nov 30 2014

nn = 77; Array[Set[a[#], # - 1] &, 2]; Do[Set[a[i], If[EvenQ[i], 2 a[i/2], a[# + 1] + a[#] &[(i - 1)/2]]], {i, 3, nn}]; Array[a, nn] (* Michael De Vlieger, Mar 22 2022 *)

a(n) = my(i=logint(n,2)-1); if(bittest(n,i), n - 2<Kevin Ryde, Aug 19 2022

from functools import lru_cache
@lru_cache(maxsize=None)
def A060973(n): return n-1 if n <= 2 else A060973(n//2) + A060973((n+1)//2) # Chai Wah Wu, Mar 08 2022

a(n) = n - A006165(n) = A006165(n) - A053646(n) = (n - A053646(n))/2 [for n > 1].
If n = 2*(2^m) + k with 0 <= k <= 2^m, then a(n) = 2^m; if n = 3*(2^m) + k with 0 <= k <= 2^m, then a(n) = 2^m + k.
G.f.: -x/(1 - x) + x/(1 - x)^2 * ( 1 + Sum_{k >= 0} t^2*(t - 1) ), t = x^(2^k). - Ralf Stephan, Sep 12 2003
Conjectures from Peter Bala, Aug 03 2022: (Start)
a(n - a(n)) = a(n - a(n - a(n - a(n)))).
If b(n) = a(a(n)) then b(n - b(n)) = b(n - b(n - b(n - b(n)))) for n >= 2. (End)
Sum_{n>=2} 1/a(n)^2 = Pi^2/6 + 2. - Amiram Eldar, Sep 08 2022

A303545 For any n > 0 and prime number p, let d_p(n) be the distance from n to the nearest p-smooth number; a(n) = Sum_{i prime} d_i(n).

Original entry on oeis.org

0, 0, 1, 0, 2, 2, 3, 0, 1, 3, 6, 4, 7, 5, 2, 0, 6, 2, 9, 6, 9, 11, 14, 8, 8, 10, 5, 6, 12, 4, 10, 0, 4, 9, 5, 4, 15, 16, 13, 12, 24, 18, 28, 18, 16, 22, 28, 16, 17, 16, 20, 20, 25, 10, 12, 12, 17, 22, 24, 8, 21, 13, 3, 0, 5, 8, 26, 18, 16, 10, 25, 8, 28, 21

Offset: 1

Rémy Sigrist, Apr 26 2018

For any n > 0 and prime number p >= A006530(n), d_p(n) = 0; hence the series in the name contains only finitely many nonzero terms and is well defined.

See also A303548 for a similar sequence.

			For n = 42:
- d_2(42) = |42 - 32| = 10,
- d_3(42) = |42 - 36| = |42 - 48| = 6,
- d_5(42) = |42 - 40| = 2,
- d_p(42) = 0 for any prime number p >= 7,
- hence a(42) = 10 + 6 + 2 = 18.

Rémy Sigrist, Table of n, a(n) for n = 1..32768
Rémy Sigrist, Colored pin plot of the first 3 * 512 terms (where the color is function of the prime p in the term d_p(n))
Index entries for sequences related to distance to nearest element of some set

Cf. A006530, A053646, A301574, A303548.

gpf(n) = if (n==1, 1, my (f=factor(n)); f[#f~, 1])
a(n) = my (v=0, pi=primepi(gpf(n))); for (d=0, oo, my (o=min(primepi(gpf(n-d)), primepi(gpf(n+d)))); if (o

a(n) = 0 iff n is a power of 2.
a(2 * n) <= 2 * a(n).
a(n) >= A053646(n) + A301574(n) (as d_2 = A053646 and d_3 = A301574).

A296239 a(n) = distance from n to nearest Fibonacci number.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 1, 1, 0, 1, 2, 2, 1, 0, 1, 2, 3, 4, 3, 2, 1, 0, 1, 2, 3, 4, 5, 6, 6, 5, 4, 3, 2, 1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 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, 16, 15, 14, 13, 12, 11, 10, 9

Offset: 0

Rémy Sigrist, Dec 09 2017

The Fibonacci numbers correspond to sequence A000045.
This sequence is analogous to:
- A051699 (distance to nearest prime),
- A053188 (distance to nearest square),
- A053646 (distance to nearest power of 2),
- A053615 (distance to nearest oblong number),
- A053616 (distance to nearest triangular number),
- A061670 (distance to nearest power),
- A074989 (distance to nearest cube),
- A081134 (distance to nearest power of 3),
The local maxima of the sequence correspond to positive terms of A004695.
a(n) = 0 iff n = A000045(k) for some k >= 0.
a(n) = 1 iff n = A061489(k) for some k > 4.
For any n >= 0, abs(a(n+1) - a(n)) <= 1.
For any n > 0, a(n) < n, and a^k(n) = 0 for some k > 0 (where a^k denotes the k-th iterate of a); k equals A105446(n) for n = 1..80 (and possibly more values).
a(n) > max(a(n-1), a(n+1)) iff n = A001076(k) for some k > 1.

			For n = 42:
- A000045(9) = 34 <= 42 <= 55 = A000045(10),
- a(42) = min(42 - 34, 55 - 42) = min(8, 13) = 8.

Cf. A000045, A001076, A004695, A051699, A053188, A053615, A053616, A053646, A061489, A061670, A074989, A081134, A105446.

fibPi[n_] := 1 + Floor[ Log[ GoldenRatio, 1 + n*Sqrt@5]]; f[n_] := Block[{m = fibPi@ n}, Min[n - Fibonacci[m -1], Fibonacci[m] - n]]; Array[f, 81, 0] (* Robert G. Wilson v, Dec 11 2017 *)
With[{nn=80,fibs=Fibonacci[Range[0,20]]},Table[Abs[n-Nearest[fibs,n]][[1]],{n,0,nn}]] (* Harvey P. Dale, Jul 02 2022 *)

a(n) = for (i=1, oo, if (n<=fibonacci(i), return (min(n-fibonacci(i-1), fibonacci(i)-n))))

a(n) = abs(n - Fibonacci(floor(log(sqrt(20)*n)/log((1 + sqrt(5))/2)-1))). - Jon E. Schoenfield, Dec 14 2017

A301574 a(n) = distance from n to nearest 3-smooth number (A003586).

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 1, 0, 0, 1, 1, 0, 1, 2, 1, 0, 1, 0, 1, 2, 3, 2, 1, 0, 1, 1, 0, 1, 2, 2, 1, 0, 1, 2, 1, 0, 1, 2, 3, 4, 5, 6, 5, 4, 3, 2, 1, 0, 1, 2, 3, 2, 1, 0, 1, 2, 3, 4, 5, 4, 3, 2, 1, 0, 1, 2, 3, 4, 3, 2, 1, 0, 1, 2, 3, 4, 4, 3, 2, 1, 0, 1, 2, 3, 4, 5, 6

Offset: 1

Altug Alkan and Rémy Sigrist, Mar 23 2018

This sequence is unbounded.

A053646 is the corresponding sequence for 2-smooth numbers (A000079).

			a(20) = a(22) = 2 because 18 is the nearest 3-smooth number to 20 and 24 is the nearest 3-smooth number to 22.

Altug Alkan, Table of n, a(n) for n = 1..10000
Rémy Sigrist, PARI program for A301574
Index entries for sequences related to distance to nearest element of some set

Cf. A000079, A003586, A053646.

PARI
```
\\ See Links section.
```

from sympy import integer_log
def A301574(n):
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def f(x): return x-sum((x//3**i).bit_length() for i in range(integer_log(x,3)[0]+1))
    k = n-f(n)
    return min(n-bisection(lambda x:f(x)+k,k,k),bisection(lambda x:f(x)+k+1,n,n)-n) # Chai Wah Wu, Oct 22 2024

a(n) = 0 iff n belongs to A003586.
2 * a(n) >= a(2 * n).
3 * a(n) >= a(3 * n).

A303548 For any n > 0 and h > 0, let d_h(n) be the distance from n to the nearest number with Hamming weight at most h; a(n) = Sum_{i > 0} d_i(n).

Original entry on oeis.org

0, 0, 1, 0, 1, 2, 2, 0, 1, 2, 4, 4, 4, 4, 3, 0, 1, 2, 4, 4, 6, 8, 9, 8, 8, 8, 9, 8, 7, 6, 4, 0, 1, 2, 4, 4, 6, 8, 9, 8, 10, 12, 15, 16, 17, 18, 18, 16, 16, 16, 17, 16, 17, 18, 18, 16, 15, 14, 14, 12, 10, 8, 5, 0, 1, 2, 4, 4, 6, 8, 9, 8, 10, 12, 15, 16, 17, 18

Offset: 1

Rémy Sigrist, Apr 26 2018

For any n > 0 and h >= A000120(n), d_h(n) = 0, hence the series in the name contains only finitely many nonzero terms and is well defined.

See also A303545 for a similar sequence.

			For n = 42:
- d_1(n) = |42 - 32| = 10,
- d_2(n) = |42 - 40| = 2,
- d_h(n) = 0 for any h >= 3,
- hence a(42) = 10 + 2 = 12.

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Rémy Sigrist, Colored pin plot of the first 3 * 512 terms (where the color is function of the number h in the term d_h(n))
Index entries for sequences related to distance to nearest element of some set

Cf. A000120, A053646, A303545.

a(n) = my (v=0, h=hamming weight(n)); for (d=0, oo, my (o=min(hamming weight(n-d), hamming weight(n+d))); if (o

a(n) = 0 iff n is a power of 2.
Apparently, a(2 * n) = 2 * a(n).
a(n) >= A053646(n) (as d_1 = A053646).

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A081252 Partial sums of A053646.

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

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A005942 a(2n) = a(n) + a(n+1), a(2n+1) = 2a(n+1), if n >= 2.

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

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A081134 Distance to nearest power of 3.

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A060973 a(2*n+1) = a(n+1)+a(n), a(2*n) = 2*a(n), with a(1)=0 and a(2)=1.

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A303545 For any n > 0 and prime number p, let d_p(n) be the distance from n to the nearest p-smooth number; a(n) = Sum_{i prime} d_i(n).

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A060973 a(2n+1) = a(n+1)+a(n), a(2n) = 2*a(n), with a(1)=0 and a(2)=1.