A053646 -id:A053646 - cp's OEIS frontend

A357562 a(n) = n - 2*b(b(n)) for n >= 2, where b(n) = A356988(n).

0, 1, 0, 1, 0, 1, 2, 1, 0, 1, 2, 3, 2, 1, 0, 1, 2, 3, 4, 5, 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, 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, 20, 19, 18, 17, 16, 15, 14, 13, 12

Offset: 2

Peter Bala, Oct 14 2022

a(n+1) - a(n) is equal to 1 or -1.
The sequence vanishes at abscissa values n = 2, 4, 6, 10, 16, 26, ..., 2*Fibonacci(k), .... For k >= 2, the line graph of the sequence, starting from the zero value at abscissa n = 2*Fibonacci(k), ascends with slope 1 to a local peak at height Fibonacci(k-1) at abscissa value n = Fibonacci(k+2) before descending with slope -1 to the next zero at abscissa n = 2*Fibonacci(k+1).
a(n) = the distance to the nearest number of the form 2*Fibonacci(k). Cf. A053646.

Peter Bala, Notes on A357562

Cf. A000045, A053646, A356988, A357563, A357564.

# b(n) = A356988(n)
b := proc(n) option remember; if n = 1 then 1 else n - b(b(n - b(b(b(n-1))))) end if; end proc:
seq( n - 2*b(b(n)), n = 2..100);

For k >= 2 there holds
a(2*Fibonacci(k) + j ) = j for 0 <= j <= Fibonacci(k-1) and
a(Fibonacci(k+2) + j) = Fibonacci(k-1) - j for 0 <= j <= Fibonacci(k-1).

A342872 Distance to nearest product of 3 consecutive numbers (three-dimensional promic number, A007531).

Original entry on oeis.org

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, 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, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17

Offset: 0

Lamine Ngom, Mar 28 2021

			a(13) = 7 since 6 is the closest three-dimensional promic to 13 and 13 - 6 = 7.

Lamine Ngom, Table of n, a(n) for n = 0..10000

Cf. A007531, A342873.

Other distance to nearest: A081134, A053646, A201053.

def a(n): return min(abs(n-k*(k+1)*(k+2)) for k in range(int(n**1/3)+1))
print([a(n) for n in range(78)]) # Michael S. Branicky, Mar 28 2021

A080776 Oscillating sequence which rises to 2^(k-1) in k-th segment (k>=1) then falls back to 0.

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, 26, 27, 28, 29, 30, 31, 32, 31, 30

Offset: 0

N. J. A. Sloane, Mar 11 2003

nonn

k-th segment has length 2^k (k>=0).

Hsien-Kuei Hwang, S Janson, TH 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; http://140.109.74.92/hk/wp-content/files/2016/12/aat-hhrr-1.pdf. Also 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

R. Stephan, Some divide-and-conquer sequences ...
R. Stephan, Table of generating functions

Essentially the same as A053646.

G.f.: -1 + 2/(1-x) + 1/(1-x)^2 * (-1 + sum(k>=0, 2t^2(t-1), t=x^2^k)). a(n) = A005942(n+2) - 3(n+1), n>0. - Ralf Stephan, Sep 13 2003

a(0)=0, a(2n) = a(n) + a(n-1) + (n==1), a(2n+1) = 2a(n). - Ralf Stephan, Oct 20 2003

A302558 For any n > 0 and m > 1, let d_m(n) be the distance from n to the nearest power of a number <= m (i.e., the distance to the nearest number of the form x^k with x <= m and k >= 0); a(n) = Sum_{i > 1} d_i(n).

Original entry on oeis.org

0, 0, 1, 0, 3, 7, 5, 0, 1, 9, 18, 28, 30, 23, 13, 0, 15, 31, 48, 66, 73, 64, 50, 33, 11, 29, 5, 29, 54, 55, 29, 0, 31, 63, 41, 16, 51, 87, 124, 162, 201, 241, 252, 231, 207, 180, 150, 117, 73, 113, 152, 192, 233, 275, 318, 362, 364, 321, 275, 226, 174, 119, 61

Offset: 1

Rémy Sigrist, Aug 15 2018

nonn

For any n > 1 and m >= n, d_m(n) = 0, hence the series in the name contains only finitely many nonzero terms and is well defined.
The set of local minima (i.e., indices n > 1 where a(n) < min(a(n-1), a(n+1))) seem to correspond to A001597 minus {1, 9}.
See A303545 for a similar sequence.

			For n = 10:
- d_2(10) = |10 - 8| = 2,
- d_m(10) = |10 - 9| = 1 for m = 3..9,
- d_m(10) = 0 for any m >= 10,
- hence a(10) = 2 + 7*1 = 9.

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Rémy Sigrist, Colored logarithmic pin plot of the first 1024 terms (where the color is function of the number m in the term d_m(n))
Index entries for sequences related to distance to nearest element of some set

Cf. A001597, A053646, A303545.

a(n) = my (v=0, d=oo); for (m=2, oo, my (k=logint(n,m)); d = min(d, min(n-m^k, m^(k+1)-n)); if (d, v+=d, return (v)))

a(n) = 0 iff n is a power of 2.

a(n) >= A053646(n) (as d_2 = A053646).

A302721 Square array T(n, k) read by antidiagonals upwards, n > 0 and k > 0: T(n, k) is the distance from n to the nearest prime(k)-smooth number (where prime(k) denotes the k-th prime number).

Original entry on oeis.org

0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 2, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3, 0, 1, 0, 0, 0, 0, 0, 0

Offset: 1

Rémy Sigrist, Apr 29 2018

			Array T(n, k) begins:
  n\k|  1  2  3  4  5  6  7  8  9 10 11 12 13 14 15 16 17 18 19 20
  ---+------------------------------------------------------------
    1|  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0
    2|  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0
    3|  1  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0
    4|  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0
    5|  1  1  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0
    6|  2  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0
    7|  1  1  1  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0
    8|  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0
    9|  1  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0
   10|  2  1  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0
   11|  3  1  1  1  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0
   12|  4  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0
   13|  3  1  1  1  1  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0

Index entries for sequences related to distance to nearest element of some set

Cf. A053646 (first column), A061395, A301574 (second column), A303545 (row sums).

gpf(n) = if (n==1, 1, my (f=factor(n)); f[#f~, 1])
T(n,k) = my (p=prime(k)); for (d=0, oo, if (gpf(n-d) <= p || gpf(n+d) <= p, return (d)))

a(2^i, k) = 0 for any i >= 0.
a(2*n, k) <= 2*a(n, k).
a(n, k+1) <= a(n, k).
abs(T(n+1, k) - T(n, k)) <= 1.
a(n, A061395(n)) = 0 for any n > 1.
a(n, 1) = A053646(n).
a(n, 2) = A301574(n).
Sum_{k > 0} a(n, k) = A303545(n).

cp's OEIS Frontend

A357562 a(n) = n - 2*b(b(n)) for n >= 2, where b(n) = A356988(n).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Formula

A342872 Distance to nearest product of 3 consecutive numbers (three-dimensional promic number, A007531).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Python

A080776 Oscillating sequence which rises to 2^(k-1) in k-th segment (k>=1) then falls back to 0.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Formula

A302558 For any n > 0 and m > 1, let d_m(n) be the distance from n to the nearest power of a number <= m (i.e., the distance to the nearest number of the form x^k with x <= m and k >= 0); a(n) = Sum_{i > 1} d_i(n).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Formula

A302721 Square array T(n, k) read by antidiagonals upwards, n > 0 and k > 0: T(n, k) is the distance from n to the nearest prime(k)-smooth number (where prime(k) denotes the k-th prime number).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

PARI

Formula