A053615 Pyramidal sequence: distance to nearest product of two consecutive integers (promic or heteromecic numbers).

0, 1, 0, 1, 2, 1, 0, 1, 2, 3, 2, 1, 0, 1, 2, 3, 4, 3, 2, 1, 0, 1, 2, 3, 4, 5, 4, 3, 2, 1, 0, 1, 2, 3, 4, 5, 6, 5, 4, 3, 2, 1, 0, 1, 2, 3, 4, 5, 6, 7, 6, 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, 8, 7, 6, 5, 4, 3, 2, 1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 9, 8, 7

Offset

0,5

Comments

a(A002378(n)) = 0; a(n^2) = n.

Table A049581 T(n,k) = |n-k| read by sides of squares from T(1,n) to T(n,n), then from T(n,n) to T(n,1). - Boris Putievskiy, Jan 29 2013

Links

Harvey P. Dale, Table of n, a(n) for n = 0..1000

Boris Putievskiy, Transformations [of] Integer Sequences And Pairing Functions arXiv:1212.2732 [math.CO], 2012.

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

Formula

a(n) = A004738(n+1) - 1.

Let u(1)=1, u(n) = n - u(n-sqrtint(n)) (cf. A037458); then a(0)=0 and for n > 0 a(n) = 2*u(n) - n. - Benoit Cloitre, Dec 22 2002

a(0)=0 then a(n) = floor(sqrt(n)) - a(n - floor(sqrt(n))). - Benoit Cloitre, May 03 2004

a(n) = |A196199(n)|. a(n) = |n - t^2 - t|, where t = floor(sqrt(n)). - Boris Putievskiy, Jan 29 2013 [corrected by Ridouane Oudra, May 11 2019]

a(n) = A000194(n) - A053188(n) = t - |t^2 - n|, where t = floor(sqrt(n)+1/2). - Ridouane Oudra, May 11 2019

Example

a(10) = |10 - 3*4| = 2.
From Boris Putievskiy, Jan 29 2013: (Start)
The start of the sequence as table:
  0, 1, 2, 3, 4, 5, 6, 7, ...
  1, 0, 1, 2, 3, 4, 5, 6, ...
  2, 1, 0, 1, 2, 3, 4, 5, ...
  3, 2, 1, 0, 1, 2, 3, 4, ...
  4, 3, 2, 1, 0, 1, 2, 3, ...
  5, 4, 3, 2, 1, 0, 1, 2, ...
  6, 5, 4, 3, 2, 1, 0, 1, ...
  ...
The start of the sequence as triangle array read by rows:
  0;
  1, 0, 1;
  2, 1, 0, 1, 2;
  3, 2, 1, 0, 1, 2, 3;
  4, 3, 2, 1, 0, 1, 2, 3, 4;
  5, 4, 3, 2, 1, 0, 1, 2, 3, 4, 5;
  6, 5, 4, 3, 2, 1, 0, 1, 2, 3, 4, 5, 6;
  7, 6, 5, 4, 3, 2, 1, 0, 1, 2, 3, 4, 5, 6, 7;
  ...
Row number r contains 2*r-1 numbers: r-1, r-2, ..., 0, 1, 2, ..., r-1. (End)

Maple

A053615 := proc(n)
    A004738(n+1)-1 ; # reuses code of A004738
end proc:
seq(A053615(n), n=0..30) ; # R. J. Mathar, Feb 14 2019

Mathematica

a[0] = 0; a[n_] := Floor[Sqrt[n]] - a[n - Floor[Sqrt[n]]]; Table[a[n], {n, 0, 103}] (* Jean-François Alcover, Dec 16 2011, after Benoit Cloitre *)
Join[{0}, Module[{nn=150, ptci}, ptci=Times@@@Partition[Range[nn/2+1], 2, 1]; Table[Abs[n-Nearest[ptci, n]], {n, nn}][[All, 1]]]] (* Harvey P. Dale, Aug 29 2020 *)

Prog

(PARI) a(n)=sqrtint(n)-a(n-sqrtint(n))
(PARI) apply( {A053615(n)=(t=sqrt(n)\/1)-abs(t^2-n)}, [0..99]) \\ M. F. Hasler, Feb 01 2025
(Python) A053615 = lambda n: (t := round(n**.5)) - abs(t**2 - n) # M. F. Hasler, Feb 01 2025

Crossrefs

Cf. A002262, A002378, A004738, A049581, A053188, A196199.

Sequence in context: A329116 A255175 A196199 * A002819 A357562 A307672

Adjacent sequences: A053612 A053613 A053614 * A053616 A053617 A053618

Keyword

easy,nice,nonn,changed

Author

Henry Bottomley, Mar 20 2000

Status

approved