A053186 -id:A053186 - cp's OEIS frontend

A176864 Numbers k such that A053186(k) is prime.

3, 6, 7, 11, 12, 14, 18, 19, 21, 23, 27, 28, 30, 32, 38, 39, 41, 43, 47, 51, 52, 54, 56, 60, 62, 66, 67, 69, 71, 75, 77, 83, 84, 86, 88, 92, 94, 98, 102, 103, 105, 107, 111, 113, 117, 119, 123, 124, 126, 128, 132, 134, 138, 140, 146, 147, 149, 151, 155, 157, 161, 163

Offset: 1

Vladimir Joseph Stephan Orlovsky, Apr 27 2010

nonn

Numbers of the form p + j^2 where p is prime and p <= 2*j. - Robert Israel, Dec 14 2018

			3-1^2=2, 6-2^2=2, 7-2^2=3, 11-3^2=2, ...

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A053186.

N:= 30: # to get all terms < (N+1)^2
Primes:= select(isprime, [2,seq(i,i=3..2*N,2)]):
seq(op(map(`+`, select(`<=`,Primes, 2*j),j^2)),j=1..N); # Robert Israel, Dec 14 2018

lst={};Do[p=n-Floor[Sqrt[n]]^2;If[PrimeQ[p],AppendTo[lst,n]],{n,6!}];lst

A010052 Characteristic function of squares: a(n) = 1 if n is a square, otherwise 0.

Original entry on oeis.org

1, 1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 0

N. J. A. Sloane

Also parity of the divisor function A000005 if n >= 1. - Omar E. Pol, Jan 14 2012
This sequence can be considered as k=1 analog of A025426 (k=2), A025427 (k=3), A025428 (k=4); see also A000161. - M. F. Hasler, Jan 25 2013
Also, the decimal expansion of Sum_{n >= 0} 1/(10^n)^n. -  Eric Desbiaux, Mar 15 2009, rephrased and simplified by M. F. Hasler, Jan 26 2013
Run lengths of zeros gives A005843, the nonnegative even numbers. - Jeremy Gardiner, Jan 14 2018
Inverse Möbius transform of Liouville's lambda function (A008836), n >= 1. - Wesley Ivan Hurt, Jun 22 2024

			G.f. = 1 + x + x^4 + x^9 + x^16 + x^25 + x^36 + x^49 + x^64 + x^81 + ...

Jean-Paul Allouche and Jeffrey Shallit, Automatic Sequences, Cambridge Univ. Press, 2003, pp. 3-4, also p. 166, Ex. 5.5.1.
Tom M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, page 48, Problem 20.
Richard Bellman, A Brief Introduction to Theta Functions, Dover, 2013 (11.14).
Michael D. Hirschhorn, The Power of q, Springer, 2017. See phi(q) page 8.
Michel Rigo, Formal Languages, Automata and Numeration Systems, 2 vols., Wiley, 2014. Mentions this sequence - see "List of Sequences" in Vol. 2.
Stephen Wolfram, A New Kind of Science, Wolfram Media, 2002, p. 55.

Charles R Greathouse IV, Table of n, a(n) for n = 0..10000
David Christopher and Tamil Nadu, Partitions with Fixed Number of Sizes, Journal of Integer Sequences, 15 (2015), Article 15.11.5.
Robert Price, Comments on A010052 concerning Elementary Cellular Automata, Jan 29 2016.
Yash Puri and Thomas Ward, Arithmetic and growth of periodic orbits, J. Integer Seqs., Vol. 4 (2001), Article 01.2.1.
Eric Weisstein's World of Mathematics, Elementary Cellular Automaton.
Eric Weisstein's World of Mathematics, Jacobi Theta Functions.
Stephen Wolfram, A New Kind of Science.
Index entries for characteristic functions
Index to Elementary Cellular Automata.
Index entries for sequences computed from exponents in factorization of n.
Index entries for sequences related to cellular automata.

Column k=1 of A243148, A337165, A341040 (for n>0).
Cf. A000005, A000122, A005369, A007913, A008836 (Mobius transf.), A037011, A063524, A258998, A271102 (Dirichlet inv), A046951 (inv. Mobius trans.).
First differences of A000196.

a010052 n = fromEnum $ a000196 n ^ 2 == n
-- Reinhard Zumkeller, Jan 26 2012, Feb 20 2011
a010052_list = concat (iterate (\xs -> xs ++ [0,0]) [1])
-- Reinhard Zumkeller, Apr 27 2012

readlib(issqr): f := i->if issqr(i) then 1 else 0; fi; [ seq(f(i),i=0..100) ];

lst = {}; Do[AppendTo[lst, 2*Sum[Floor[n/k] - Floor[(n - 1)/k], {k, Floor[Sqrt[n]]}] - DivisorSigma[0, n]], {n, 93}]; Prepend[lst, 1] (* Eric Desbiaux, Jan 29 2012 *)
Table[If[IntegerQ[Sqrt[n]],1,0],{n,0,100}] (* Harvey P. Dale, Jul 19 2014 *)
a[n_] := SeriesCoefficient[1/(1 - q)* QHypergeometricPFQ[{-q, -q}, {-(q^2)}, -q, -q], {q, 0, Abs@n}] (* Mats Granvik, Jan 01 2016 *)
Range[0, 120] /. {n_ /; IntegerQ@ Sqrt@ n -> 1, n_ /; n != 1 -> 0} (* Michael De Vlieger, Jan 02 2016 *)
a[n_] := Sum[If[Mod[n, k] == 0, Re[Sqrt[LiouvilleLambda[k]]*Sqrt[LiouvilleLambda[n/k]]], 0], {k, 1, n}] (* Mats Granvik, Aug 10 2018 *)

PARI
```
{a(n) = issquare(n)};
```

a(n)=if(n<1,1,sumdiv(n,d,(-1)^bigomega(d))) \\ Benoit Cloitre, Oct 25 2009

a(n) = if (n<1, 1, direuler( p=2, n, 1/ (1 - X^2 ))[n]); \\ Michel Marcus, Mar 08 2015

def A010052(n): return int(math.isqrt(n)**2==n) ##  appears to be faster than sympy.ntheory.primetest.is_square, up to 10^8 at least.
# M. F. Hasler, Mar 21 2022

(define (A010052 n) (if (zero? n) 1 (- (A000196 n) (A000196 (- n 1))))) ;; (For the definition of A000196, see under that entry). - Antti Karttunen, Nov 03 2017

a(n) = floor(sqrt(n)) - floor(sqrt(n-1)), for n > 0.
a(n) = A000005(n) mod 2, n > 0. - Ahmed Fares (ahmedfares(AT)my-deja.com), Apr 19 2001
G.f. A(x) satisfies: 0 = f(A(x), A(x^2), A(x^4)) where f(u, v, w) = (u-w)^2 - (v-w)*(v+w-1) - Michael Somos, Jul 19 2004
Dirichlet g.f.: zeta(2s). - Franklin T. Adams-Watters, Sep 11 2005
G.f.: (theta_3(0,x) + 1)/2, where theta_3 is a Jacobi theta function. - Franklin T. Adams-Watters, Jun 19 2006 [See A000122 for theta_3.]
a(n) = f(n,0) with f(x,y) = f(x-2*y-1,y+1) if x > 0, otherwise 0^(-x). - Reinhard Zumkeller, Sep 26 2008
a(n) = Sum_{d|n} (-1)^bigomega(d), for n >= 1. - Benoit Cloitre, Oct 25 2009
a(n) <= A093709(n). - Reinhard Zumkeller, Nov 14 2009
a(A000290(n)) = 1; a(A000037(n)) = 0. - Reinhard Zumkeller, Jun 20 2011
a(n) = 0 ^ A053186(n). - Reinhard Zumkeller, Feb 12 2012
a(n) = A063524(A007913(n)), for n > 0. - Reinhard Zumkeller, Jul 09 2014
a(n) = -(-1)^n * A258998(n) unless n = 0. 2 * a(n) = A000122(n) unless n = 0. - Michael Somos, Jun 16 2015
a(n) = A037011(A156552(n)), provided that A037011(n) = A000035(A106737(n)). [See A037011.] - Antti Karttunen, Nov 03 2017
a(n*m) = a(n/gcd(n,m))*a(m/gcd(n,m)) for all n and m > 0 (conjectured). - Velin Yanev, Feb 13 2019 [Proof from Michael B. Porter, Feb 16 2019: If nm is a square, nm = product_i (p_i^2), where p_i are prime, not necessarily distinct. Each p_i either appears twice in n, twice in m, or one time in each and therefore in the gcd. So n/gcd(n,m) and m/gcd(n,m) are both squares. If nm is not a square, there is a q_j that appears in one of n or m but not in the gcd. So either n/gcd(n,m) or m/gcd(n,m) is not a square.]
a(n) = Sum_{d|n} A008836(d), n >= 1, a(0) = 1. - Jinyuan Wang, Apr 20 2019
G.f.: A(q) = Sum_{n >= 0}  q^(2*n)*Product_{k >= 2*n+1} 1 - (-q)^k. - Peter Bala, Feb 22 2021
Multiplicative with a(p^e) = 1 if e is even, and 0 otherwise. - Amiram Eldar, Dec 29 2022
a(n) = Sum_{d|n} mobius(core(n)), where core(n) = A007913(n). - Peter Bala, Jan 24 2024

More terms from Franklin T. Adams-Watters, Jun 19 2006

A000196 Integer part of square root of n. Or, number of positive squares <= n. Or, n appears 2n+1 times.

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane

Also the integer part of the geometric mean of the divisors of n. - Amarnath Murthy, Dec 19 2001
Number of numbers k (<= n) with an odd number of divisors. - Benoit Cloitre, Sep 07 2002
Also, for n > 0, the number of digits when writing n in base where place values are squares, cf. A007961; A190321(n) <= a(n). - Reinhard Zumkeller, May 08 2011
The least monotonic left inverse of squares, A000290. That is, the lexicographically least nondecreasing sequence a(n) such that a(A000290(n)) = n. - Antti Karttunen, Oct 06 2017

			G.f. = x + x^2 + x^3 + 2*x^4 + 2*x^5 + 2*x^6 + 2*x^7 + 2*x^8 + 3*x^9 + ...

Tom M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, p. 73, problem 23.
Lionel Levine, Fractal sequences and restricted Nim, Ars Combin. 80 (2006), 113-127.
Paul J. McCarthy, Introduction to Arithmetical Functions, Springer Verlag, 1986, p. 28.
N. J. A. Sloane and Allan Wilks, On sequences of Recaman type, paper in preparation, 2006.

Franklin T. Adams-Watters, Table of n, a(n) for n = 0..10000
Krassimir Atanassov, On Some of Smarandache's Problems
Krassimir Atanassov, On the 100-th, the 101-st and 102-nd Smarandache's problems, Notes on Number Theory and Discrete Mathematics, Sophia, Bulgaria, Vol. 5 (1999), No. 3, 94-96.
Henry Bottomley, Illustration of A000196, A048760, A053186.
Matthew Hyatt and Marina Skyers, On the Increases of the Sequence floor(k*sqrt(n)), Electronic Journal of Combinatorial Number Theory, Volume 15 (2015), #A17.
Lionel Levine, Fractal sequences and restricted Nim, arXiv:math/0409408 [math.CO], 2004.
Paul Pollack and Joseph Vandehey, Besicovitch, Bisection, and the Normality of 0.(1)(4)(9)(16)(25)...., The American Mathematical Monthly 122.8 (2015): 757-765.
N. J. A. Sloane, Families of Essentially Identical Sequences, Mar 24 2021 (Includes this sequence)
Florentin Smarandache, Only Problems, Not Solutions!, 1993.

Cf. A000290, A002162, A003056, A028391, A048760, A048766, A074704, A079051.

Column k=1 of A281871.

import Data.Bits (shiftL, shiftR)
a000196 :: Integer -> Integer
a000196 0 = 0
a000196 n = newton n (findx0 n 1) where
   -- find x0 == 2^(a+1), such that 4^a <= n < 4^(a+1).
   findx0 0 b = b
   findx0 a b = findx0 (a `shiftR` 2) (b `shiftL` 1)
   newton n x = if x' < x then newton n x' else x
                where x' = (x + n `div` x) `div` 2
a000196_list = concat $ zipWith replicate [1,3..] [0..]
-- Reinhard Zumkeller, Apr 12 2012, Oct 23 2010

a(n) = isqrt(n) # Paul Muljadi, Jun 03 2024

Magma
```
[Isqrt(n) : n in [0..100]];
```

Digits := 100; A000196 := n->floor(evalf(sqrt(n)));

Table[n, {n, 0, 20}, {2n + 1}] //Flatten (* Zak Seidov Mar 19 2011 *)
IntegerPart[Sqrt[Range[0, 110]]] (* Harvey P. Dale, May 23 2012 *)
Floor[Sqrt[Range[0, 99]]] (* Alonso del Arte, Dec 31 2013 *)
a[ n_] := SeriesCoefficient[ (EllipticTheta[ 3, 0, x]  - 1) / (2 (1 - x)), {x, 0, n}]; (* Michael Somos, May 28 2014 *)

PARI
```
{a(n) = if( n<0, 0, sqrtint(n))};
```

# from http://code.activestate.com/recipes/577821-integer-square-root-function/
def A000196(n):
  if n < 0:
    raise ValueError('only defined for nonnegative n')
  if n == 0:
    return 0
  a, b = divmod(n.bit_length(), 2)
  j = 2**(a+b)
  while True:
    k = (j + n//j)//2
    if k >= j:
      return j
    j = k
print([A000196(n)for n in range(102)])
# Jason Kimberley, Nov 09 2016

from math import isqrt
def a(n): return isqrt(n)
print([a(n) for n in range(102)]) # Michael S. Branicky, Feb 15 2023

;; The following implementation uses higher order function LEFTINV-LEASTMONO-NC2NC from my IntSeq-library. It returns the least monotonic left inverse of any strictly growing function (see the comment-section for the definition) and although it does not converge as fast to the result as many specialized integer square root algorithms, at least it does not involve any floating point arithmetic. Thus with correctly implemented bignums it will produce correct results even with very large arguments, in contrast to just taking the floor of (sqrt n).
;; Source of LEFTINV-LEASTMONO-NC2NC can be found under https://github.com/karttu/IntSeq/blob/master/src/Transforms/transforms-core.ss and the definition of A000290 is given under that entry.
(define A000196 (LEFTINV-LEASTMONO-NC2NC 0 0 A000290)) ;; Antti Karttunen, Oct 06 2017

a(n) = Card(k, 0 < k <= n such that k is relatively prime to core(k)) where core(x) is the squarefree part of x. - Benoit Cloitre, May 02 2002
a(n) = a(n-1) + floor(n/(a(n-1)+1)^2), a(0) = 0. - Reinhard Zumkeller, Apr 12 2004
From Hieronymus Fischer, May 26 2007: (Start)
a(n) = Sum_{k=1..n} A010052(k).
G.f.: g(x) = (1/(1-x))*Sum_{j>=1} x^(j^2) = (theta_3(0, x) - 1)/(2*(1-x)) where theta_3 is a Jacobi theta function. (End)
a(n) = floor(A000267(n)/2). - Reinhard Zumkeller, Jun 27 2011
a(n) = floor(sqrt(n)). - Arkadiusz Wesolowski, Jan 09 2013
Sum_{n>0} 1/a(n)^s = 2*zeta(s-1) + zeta(s), where zeta is the Riemann zeta function. - Enrique Pérez Herrero, Oct 15 2013
From Wesley Ivan Hurt, Dec 31 2013: (Start)
a(n) = Sum_{i=1..n} (A000005(i) mod 2), n > 0.
a(n) = (1/2)*Sum_{i=1..n} (1 - (-1)^A000005(i)), n > 0. (End)
a(n) = sqrt(A048760(n)), n >= 0. - Wolfdieter Lang, Mar 24 2015
a(n) = Sum_{k=1..n} floor(n/k)*lambda(k) = Sum_{m=1..n} Sum_{d|m} lambda(d) where lambda(j) is Liouville lambda function, A008836. - Geoffrey Critzer, Apr 01 2015
Sum_{n>=1} (-1)^(n+1)/a(n) = log(2) (A002162). - Amiram Eldar, May 02 2023

A005563 a(n) = n*(n+2) = (n+1)^2 - 1.

Original entry on oeis.org

0, 3, 8, 15, 24, 35, 48, 63, 80, 99, 120, 143, 168, 195, 224, 255, 288, 323, 360, 399, 440, 483, 528, 575, 624, 675, 728, 783, 840, 899, 960, 1023, 1088, 1155, 1224, 1295, 1368, 1443, 1520, 1599, 1680, 1763, 1848, 1935, 2024, 2115, 2208, 2303, 2400, 2499, 2600

Offset: 0

N. J. A. Sloane

Erdős conjectured that n^2 - 1 = k! has a solution if and only if n is 5, 11 or 71 (when k is 4, 5 or 7).
Second-order linear recurrences y(m) = 2y(m-1) + a(n)*y(m-2), y(0) = y(1) = 1, have closed form solutions involving only powers of integers. - Len Smiley, Dec 08 2001
Number of edges in the join of two cycle graphs, both of order n, C_n * C_n. - Roberto E. Martinez II, Jan 07 2002
Let k be a positive integer, M_n be the n X n matrix m_(i,j) = k^abs(i-j) then det(M_n) = (-1)^(n-1)*a(k-1)^(n-1). - Benoit Cloitre, May 28 2002
Also numbers k such that 4*k + 4 is a square. - Cino Hilliard, Dec 18 2003
For each term k, the function sqrt(x^2 + 1), starting with 1, produces an integer after k iterations. - Gerald McGarvey, Aug 19 2004
a(n) mod 3 = 0 if and only if n mod 3 > 0: a(A008585(n)) = 2; a(A001651(n)) = 0; a(n) mod 3 = 2*(1-A079978(n)). - Reinhard Zumkeller, Oct 16 2006
a(n) is the number of divisors of a(n+1) that are not greater than n. - Reinhard Zumkeller, Apr 09 2007
Nonnegative X values of solutions to the equation X^3 + X^2 = Y^2. To find Y values: b(n) = n(n+1)(n+2). - Mohamed Bouhamida, Nov 06 2007
Sequence allows us to find X values of the equation: X + (X + 1)^2 + (X + 2)^3 = Y^2. To prove that X = n^2 + 2n: Y^2 = X + (X + 1)^2 + (X + 2)^3 = X^3 + 7*X^2 + 15X + 9 = (X + 1)(X^2 + 6X + 9) = (X + 1)*(X + 3)^2 it means: (X + 1) must be a perfect square, so X = k^2 - 1 with k>=1. we can put: k = n + 1, which gives: X = n^2 + 2n and Y = (n + 1)(n^2 + 2n + 3). - Mohamed Bouhamida, Nov 12 2007
From R. K. Guy, Feb 01 2008: (Start)
Toads and Frogs puzzle:
This is also the number of moves that it takes n frogs to swap places with n toads on a strip of 2n + 1 squares (or positions, or lily pads) where a move is a single slide or jump, illustrated for n = 2, a(n) = 8 by
   T T - F F
   T - T F F
   T F T - F
   T F T F -
   T F - F T
   - F T F T
   F - T F T
   F F T - T
   F F - T T
I was alerted to this by the Holton article, but on consulting Singmaster's sources, I find that the puzzle goes back at least to 1867.
Probably the first to publish the number of moves for n of each animal was Edouard Lucas in 1883. (End)
a(n+1) = terms of rank 0, 1, 3, 6, 10 = A000217 of A120072 (3, 8, 5, 15). - Paul Curtz, Oct 28 2008
Row 3 of array A163280, n >= 1. - Omar E. Pol, Aug 08 2009
Final digit belongs to a periodic sequence: 0, 3, 8, 5, 4, 5, 8, 3, 0, 9. - Mohamed Bouhamida, Sep 04 2009 [Comment edited by N. J. A. Sloane, Sep 24 2009]
Let f(x) be a polynomial in x. Then f(x + n*f(x)) is congruent to 0 (mod f(x)); here n belongs to N. There is nothing interesting in the quotients f(x + n*f(x))/f(x) when x belongs to Z. However, when x is irrational these quotients consist of two parts, a) rational integers and b) integer multiples of x. The present sequence represents the non-integer part when the polynomial is x^2 + x + 1 and x = sqrt(2), f(x+n*f(x))/f(x) = A056108(n) + a(n)*sqrt(2). - A.K. Devaraj, Sep 18 2009
For n >= 1, a(n) is the number for which 1/a(n) = 0.0101... (A000035) in base (n+1). - Rick L. Shepherd, Sep 27 2009
For n > 0, continued fraction [n, 1, n] = (n+1)/a(n); e.g., [6, 1, 6] = 7/48. - Gary W. Adamson, Jul 15 2010
Starting (3, 8, 15, ...) = binomial transform of [3, 5, 2, 0, 0, 0, ...]; e.g., a(3) = 15 = (1*3 + 2*5 +1*2) = (3 + 10 + 2). - Gary W. Adamson, Jul 30 2010
a(n) is essentially the case 0 of the polygonal numbers. The polygonal numbers are defined as P_k(n) = Sum_{i=1..n} ((k-2)*i-(k-3)). Thus P_0(n) = 2*n-n^2 and a(n) = -P_0(n+2). See also A067998 and for the case k=1 A080956. - Peter Luschny, Jul 08 2011
a(n) is the maximal determinant of a 2 X 2 matrix with integer elements from {1, ..., n+1}, so the maximum determinant of a 2x2 matrix with integer elements from {1, ..., 5} = 5^2 - 1 = a(4) = 24. - Aldo González Lorenzo, Oct 12 2011
Using four consecutive triangular numbers t1, t2, t3 and t4, plot the points (0, 0), (t1, t2), and (t3, t4) to create a triangle. Twice the area of this triangle are the numbers in this sequence beginning with n = 1 to give 8. - J. M. Bergot, May 03 2012
Given a particle with spin S = n/2 (always a half-integer value), the quantum-mechanical expectation value of the square of the magnitude of its spin vector evaluates to  = S(S+1) = n(n+2)/4, i.e., one quarter of a(n) with n = 2S. This plays an important role in the theory of magnetism and magnetic resonance. - Stanislav Sykora, May 26 2012
Twice the harmonic mean [H(x, y) = (2*x*y)/(x + y)] of consecutive triangular numbers A000217(n) and A000217(n+1). - Raphie Frank, Sep 28 2012
Number m such that floor(sqrt(m)) = floor(m/floor(sqrt(m))) - 2 for m > 0. - Takumi Sato, Oct 10 2012
The solutions of equation 1/(i - sqrt(j)) = i + sqrt(j), when i = (n+1), j = a(n). For n = 1, 2 + sqrt(3) = 3.732050.. = A019973. For n = 2, 3 + sqrt(8) = 5.828427... = A156035. - Kival Ngaokrajang, Sep 07 2013
The integers in the closed form solution of a(n) = 2*a(n-1) + a(m-2)*a(n-2), n >= 2, a(0) = 0, a(1) = 1 mentioned by Len Smiley, Dec 08 2001, are m and -m + 2 where m >= 3 is a positive integer. - Felix P. Muga II, Mar 18 2014
Let m >= 3 be a positive integer. If a(n) = 2*a(n-1) + a(m-2) * a(n-2), n >= 2, a(0) = 0, a(1) = 1, then lim_{n->oo} a(n+1)/a(n) = m. - Felix P. Muga II, Mar 18 2014
For n >= 4 the Szeged index of the wheel graph W_n (with n + 1 vertices). In the Sarma et al. reference, Theorem 2.7 is incorrect. - Emeric Deutsch, Aug 07 2014
If P_{k}(n) is the n-th k-gonal number, then a(n) = t*P_{s}(n+2) - s*P_{t}(n+2) for s=t+1. - Bruno Berselli, Sep 04 2014
For n >= 1, a(n) is the dimension of the simple Lie algebra A_n. - Wolfdieter Lang, Oct 21 2015
Finding all positive integers (n, k) such that n^2 - 1 = k! is known as Brocard's problem, (see A085692). - David Covert, Jan 15 2016
For n > 0, a(n) mod (n+1) = a(n) / (n+1) = n. - Torlach Rush, Apr 04 2016
Conjecture: When using the Sieve of Eratosthenes and sieving (n+1..a(n)), with divisors (1..n) and n>0, there will be no more than a(n-1) composite numbers. - Fred Daniel Kline, Apr 08 2016
a(n) mod 8 is periodic with period 4 repeating (0,3,0,7), that is a(n) mod 8 = 5/2 - (5/2) cos(n*Pi) - sin(n*Pi/2) + sin(3*n*Pi/2). - Andres Cicuttin, Jun 02 2016
Also for n > 0, a(n) is the number of times that n-1 occurs among the first (n+1)! terms of A055881. - R. J. Cano, Dec 21 2016
The second diagonal of composites (the only prime is number 3) from the right on the Klauber triangle (see Kival Ngaokrajang link), which is formed by taking the positive integers and taking the first 1, the next 3, the following 5, and so on, each centered below the last. - Charles Kusniec, Jul 03 2017
Also the number of independent vertex sets in the n-barbell graph. - Eric W. Weisstein, Aug 16 2017
Interleaving of A000466 and A033996. - Bruce J. Nicholson, Nov 08 2019
a(n) is the number of degrees of freedom in a triangular cell for a Raviart-Thomas or Nédélec first kind finite element space of order n. - Matthew Scroggs, Apr 22 2020
From Muge Olucoglu, Jan 19 2021: (Start)
For n > 1, a(n-2) is the maximum number of elements in the second stage of the Quine-McCluskey algorithm whose minterms are not covered by the functions of n bits. At n=3, we have a(3-2) = a(1) = 1*(1+2) = 3 and f(A,B,C) = sigma(0,1,2,5,6,7).
.
          0 1 2   5 6 7
        +---------------
  *(0,1)| X X
   (0,2)| X   X
   (1,5)|   X     X
  *(2,6)|     X     X
  *(5,7)|         X   X
   (6,7)|           X X
.
*: represents the elements that are covered. (End)
1/a(n) is the ratio of the sum of the first k odd numbers and the sum of the next n*k odd numbers. - Melvin Peralta, Jul 15 2021
For n >= 1, the continued fraction expansion of sqrt(a(n)) is [n; {1, 2n}]. - Magus K. Chu, Sep 09 2022
Number of diagonals parallel to an edge in a regular (2*n+4)-gon (cf. A367204). - Paolo Xausa, Nov 21 2023
For n >= 1, also the number of minimum cyclic edge cuts in the (n+2)-trapezohedron graph. - Eric W. Weisstein, Nov 21 2024
For n >= 1, a(n) is the sum of the interior angles of a polygon with n+2 sides, in radians, multiplied by (n+2)/Pi. - Stuart E Anderson, Aug 06 2025

			G.f. = 3*x + 8*x^2 + 15*x^3 + 24*x^4 + 35*x^5 + 48*x^6 + 63*x^7 + 80*x^8 + ...

E. R. Berlekamp, J. H. Conway and R. K. Guy, Winning Ways, Academic Press, NY, 2 vols., 1982, see index under Toads and Frogs Puzzle.
Martin Gardner, Perplexing Puzzles and Tantalizing Teasers, p. 21 (for "The Dime and Penny Switcheroo").
R. K. Guy, Unsolved Problems in Theory of Numbers, Section D25.
Derek Holton, Math in School, 37 #1 (Jan 2008) 20-22.
Edouard Lucas, Récréations Mathématiques, Gauthier-Villars, Vol. 2 (1883) 141-143.
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
Jeremiah Bartz, Bruce Dearden, and Joel Iiams, Classes of Gap Balancing Numbers, arXiv:1810.07895 [math.NT], 2018.
Spencer Daugherty, Pamela E. Harris, Ian Klein, and Matt McClinton, Metered Parking Functions, arXiv:2406.12941 [math.CO], 2024. See p. 22.
R. K. Guy, Letter to N. J. A. Sloane, May 1990.
R. K. Guy, Catwalks, sandsteps and Pascal pyramids, J. Integer Sequences, Vol. 3 (2000), Article #00.1.6.
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.
Milan Janjic, Enumerative Formulas for Some Functions on Finite Sets.
F. P. Muga II, Extending the Golden Ratio and the Binet-de Moivre Formula, March 2014; Preprint on ResearchGate.
Kival Ngaokrajang, Klauber triangle
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992.
Luis Manuel Rivera, Integer sequences and k-commuting permutations, arXiv preprint arXiv:1406.3081 [math.CO], 2014.
K. V. S. Sarma and I. V. N. Uma, On Szeged index of standard graphs, International J. of Math. Archive, 3(8), 2012, 3129-3135. - _Emeric Deutsch_, Aug 07 2014
Stanislav Sýkora, Magnetic Resonance on OEIS, Stan's NMR Blog (Dec 31, 2014), Retrieved Nov 12, 2019.
Leo Tavares, Illustration: Bounded Squares.
Leo Tavares, Illustration: Trapezoids.
Leo Tavares, Illustration: Trapagons.
Eric Weisstein's World of Mathematics, Barbell Graph.
Eric Weisstein's World of Mathematics, Cyclic Edge Cut.
Eric Weisstein's World of Mathematics, Independent Vertex Set.
Eric Weisstein's World of Mathematics, Near-Square Prime.
Eric Weisstein's World of Mathematics, Trapezohedral Graph.
Wikipedia, Quine-McCluskey Algorithm.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

A column of triangle A102537.
a(n+1), n>=2, first column of triangle A120070.
Cf. A000035, A000217, A000290, A000466, A001651, A002378, A005563.
Cf. A007531, A008585, A008833, A013468, A016742, A019973, A028560.
Cf. A033996, A046092, A053186, A054000, A055881, A056108, A062196.
Cf. A068310, A067725, A067728, A067998, A069817, A079978, A080956.
Cf. A085692, A120072, A123865, A123866, A123867, A123868, A134582.
Cf. A140091, A140681, A143053, A144396, A156035, A163280, A212331.
Cf. A253607, A367204.

a005563 n = n * (n + 2)
a005563_list = zipWith (*) [0..] [2..]  -- Reinhard Zumkeller, Dec 16 2012

[n*(n+2): n in [0..60]]; // G. C. Greubel, Mar 29 2024

Table[n^2 - 1, {n, 42}] (* Zerinvary Lajos, Mar 21 2007 *)
ListCorrelate[{1, 2}, Range[-1, 50], {1, -1}, 0, Plus, Times] (* Harvey P. Dale, Aug 29 2015 *)
Range[20]^2 - 1 (* Eric W. Weisstein, Aug 16 2017 *)
Table[n (n + 2), {n, 20}] (* Eric W. Weisstein, Nov 21 2024 *)
CoefficientList[Series[(-3 + x)/(-1 + x)^3, {x, 0, 20}], x] (* Eric W. Weisstein, Nov 21 2024 *)
LinearRecurrence[{3, -3, 1}, {3, 8, 15}, 20] (* Eric W. Weisstein, Nov 21 2024 *)

makelist(n*(n+2), n, 0, 56); /* Martin Ettl, Oct 15 2012 */

a(n)=n*(n+2) \\ Charles R Greathouse IV, Dec 22 2011

concat(0, Vec(x*(3-x)/(1-x)^3 + O(x^90))) \\ Altug Alkan, Oct 22 2015

[n*(n+2) for n in range(61)] # G. C. Greubel, Mar 29 2024

G.f.: x*(3-x)/(1-x)^3. - Simon Plouffe in his 1992 dissertation
a(n) = A000290(n+1) - 1.
A002378(a(n)) = A002378(n)*A002378(n+1); e.g., A002378(15)=240=12*20. - Charlie Marion, Dec 29 2003
a(n) = A067725(n)/3. - Zerinvary Lajos, Mar 06 2007
a(n) = Sum_{k=1..n} A144396(k). - Zerinvary Lajos, May 11 2007
a(n) = A134582(n+1)/4. - Zerinvary Lajos, Feb 01 2008
A143053(a(n)) = A000290(n+1), for n > 0. - Reinhard Zumkeller, Jul 20 2008
a(n) = Real((n+1+i)^2). - Gerald Hillier, Oct 12 2008
A053186(a(n)) = 2*n. - Reinhard Zumkeller, May 20 2009
a(n) = (n! + (n+1)!)/(n-1)!, n > 0. - Gary Detlefs, Aug 10 2009
a(n) = floor(n^5/(n^3+1)) with offset 1 (a(1)=0). - Gary Detlefs, Feb 11 2010
a(n) = a(n-1) + 2*n + 1 (with a(0)=0). - Vincenzo Librandi, Nov 18 2010
Sum_{n>=1} 1/a(n) = 3/4. - Mohammad K. Azarian, Dec 29 2010
a(n) = 2/(Integral_{x=0..Pi/2} (sin(x))^(n-1)*(cos(x))^3), for n > 0. - Francesco Daddi, Aug 02 2011
a(n) = A002378(n) + floor(sqrt(A002378(n))); pronic number + its root. - Fred Daniel Kline, Sep 16 2011
a(n-1) = A008833(n) * A068310(n) for n > 1. - Reinhard Zumkeller, Nov 26 2011
G.f.: U(0) where U(k) = -1 + (k+1)^2/(1 - x/(x + (k+1)^2/U(k+1))); (continued fraction, 3-step). - Sergei N. Gladkovskii, Oct 19 2012
a(n) = 15*C(n+4,3)*C(n+4,5)/(C(n+4,2)*C(n+4,4)). - Gary Detlefs, Aug 05 2013
a(n) = (n+2)!/((n-1)! + n!), n > 0. - Ivan N. Ianakiev, Nov 11 2013
a(n) = 3*C(n+1,2) - C(n,2) for n >= 0. - Felix P. Muga II, Mar 11 2014
a(n) = (A016742(n+1) - 4)/4 for n >= 0. - Felix P. Muga II, Mar 11 2014
a(-2 - n) = a(n) for all n in Z. - Michael Somos, Aug 07 2014
A253607(a(n)) = 1. - Reinhard Zumkeller, Jan 05 2015
E.g.f.: x*(x + 3)*exp(x). - Ilya Gutkovskiy, Jun 03 2016
For n >= 1, a(n^2 + n - 2) = a(n-1) * a(n). - Miko Labalan, Oct 15 2017
Sum_{n>=1} (-1)^(n+1)/a(n) = 1/4. - Amiram Eldar, Nov 04 2020
From Amiram Eldar, Feb 17 2021: (Start)
Product_{n>=1} (1 + 1/a(n)) = 2.
Product_{n>=1} (1 - 1/a(n)) = -sqrt(2)*sin(sqrt(2)*Pi)/Pi. (End)
a(n) = A000290(n+2) - n*2. See Bounded Squares illustration. - Leo Tavares, Oct 05 2021
From Leo Tavares, Oct 10 2021: (Start)
a(n) = A008585(n) + 2*A000217(n-1). See Trapezoids illustration.
2*A005563 = A054000(n+1). See Trapagons illustration.
a(n) = 2*A000217(n) + n. (End)
a(n) = (n+2)!!/(n-2)!! for n > 1. - Jacob Szlachetka, Jan 02 2022

Partially edited by Joerg Arndt, Mar 11 2010

More terms from N. J. A. Sloane, Aug 01 2010

A002262 Triangle read by rows: T(n,k) = k, 0 <= k <= n, in which row n lists the first n+1 nonnegative integers.

Original entry on oeis.org

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

Offset: 0

Angele Hamel (amh(AT)maths.soton.ac.uk)

The point with coordinates (x = A025581(n), y = A002262(n)) sweeps out the first quadrant by upwards antidiagonals. N. J. A. Sloane, Jul 17 2018
Old name: Integers 0 to n followed by integers 0 to n+1 etc.
a(n) = n - the largest triangular number <= n. - Amarnath Murthy, Dec 25 2001
The PARI functions t1, t2 can be used to read a square array T(n,k) (n >= 0, k >= 0) by antidiagonals downwards: n -> T(t1(n), t2(n)). - Michael Somos, Aug 23 2002
Values x of unique solution pair (x,y) to equation T(x+y) + x = n, where T(k)=A000217(k). - Lekraj Beedassy, Aug 21 2004
a(A000217(n)) = 0; a(A000096(n)) = n. - Reinhard Zumkeller, May 20 2009
Concatenation of the set representation of ordinal numbers, where the n-th ordinal number is represented by the set of all ordinals preceding n, 0 being represented by the empty set. - Daniel Forgues, Apr 27 2011
An integer sequence is nonnegative if and only if it is a subsequence of this sequence. - Charles R Greathouse IV, Sep 21 2011
a(A195678(n)) = A000040(n) and a(m) <> A000040(n) for m < A195678(n), an example of the preceding comment. - Reinhard Zumkeller, Sep 23 2011
A sequence B is called a reluctant sequence of sequence A, if B is triangle array read by rows: row number k coincides with first k elements of the sequence A. A002262 is reluctant sequence of 0,1,2,3,... The nonnegative integers, A001477. - Boris Putievskiy, Dec 12 2012

			From _Daniel Forgues_, Apr 27 2011: (Start)
Examples of set-theoretic representation of ordinal numbers:
  0: {}
  1: {0} = {{}}
  2: {0, 1} = {0, {0}} = {{}, {{}}}
  3: {0, 1, 2} = {{}, {0}, {0, 1}} = ... = {{}, {{}}, {{}, {{}}}} (End)
From _Omar E. Pol_, Jul 15 2012: (Start)
  0;
  0, 1;
  0, 1, 2;
  0, 1, 2, 3;
  0, 1, 2, 3, 4;
  0, 1, 2, 3, 4, 5;
  0, 1, 2, 3, 4, 5, 6;
  0, 1, 2, 3, 4, 5, 6, 7;
  0, 1, 2, 3, 4, 5, 6, 7, 8;
  0, 1, 2, 3, 4, 5, 6, 7, 8, 9;
  0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10;
(End)

Charles R Greathouse IV, Rows n = 0..100, flattened
Boris Putievskiy, Transformations [Of] Integer Sequences And Pairing Functions, arXiv preprint arXiv:1212.2732 [math.CO], 2012.
Michael Somos, Sequences used for indexing triangular or square arrays

Cf. A002024, A002260, A004736, A025581, A025675, A025682.
Cf. A025691, A048645, A053186, A053645, A056558, A127324.
As a sequence, essentially same as A048151.
Cf. A060510 (parity).

a002262 n k = a002262_tabl !! n !! k
a002262_row n = a002262_tabl !! n
a002262_tabl = map (enumFromTo 0) [0..]
a002262_list = concat a002262_tabl
-- Reinhard Zumkeller, Aug 05 2015, Jul 13 2012, Mar 07 2011

seq(seq(i,i=0..n),n=0..14); # Peter Luschny, Sep 22 2011
A002262 := n -> n - binomial(floor((1/2)+sqrt(2*(1+n))),2);

m[n_]:= Floor[(-1 + Sqrt[8n - 7])/2]
b[n_]:= n - m[n] (m[n] + 1)/2
Table[m[n], {n, 1, 105}]     (* A003056 *)
Table[b[n], {n, 1, 105}]     (* A002260 *)
Table[b[n] - 1, {n, 1, 120}] (* A002262 *)
(* Clark Kimberling, Jun 14 2011 *)
Flatten[Table[k, {n, 0, 14}, {k, 0, n}]] (* Alonso del Arte, Sep 21 2011 *)
Flatten[Table[Range[0,n], {n,0,15}]] (* Harvey P. Dale, Jan 31 2015 *)

PARI
```
a(n)=n-binomial(round(sqrt(2+2*n)),2)
```

t1(n)=n-binomial(floor(1/2+sqrt(2+2*n)),2) /* A002262, this sequence */

t2(n)=binomial(floor(3/2+sqrt(2+2*n)),2)-(n+1) /* A025581, cf. comment by Somos for reading arrays by antidiagonals */

concat(vector(15,n,vector(n,i,i-1)))  \\ M. F. Hasler, Sep 21 2011

apply( {A002262(n)=n-binomial(sqrtint(8*n+8)\/2,2)}, [0..99]) \\ M. F. Hasler, Oct 20 2022

for i in range(16):
    for j in range(i):
        print(j, end=", ") # Mohammad Saleh Dinparvar, May 13 2020

from math import comb, isqrt
def a(n): return n - comb((1+isqrt(8+8*n))//2, 2)
print([a(n) for n in range(105)]) # Michael S. Branicky, May 07 2023

a(n) = A002260(n) - 1.
a(n) = n - (trinv(n)*(trinv(n)-1))/2; trinv := n -> floor((1+sqrt(1+8*n))/2) (cf. A002024); # gives integral inverses of triangular numbers
a(n) = n - A000217(A003056(n)) = n - A057944(n). - Lekraj Beedassy, Aug 21 2004
a(n) = A140129(A023758(n+2)). - Reinhard Zumkeller, May 14 2008
a(n) = f(n,1) with f(n,m) = if nReinhard Zumkeller, May 20 2009
a(n) = (1/2)*(t - t^2 + 2*n), where t = floor(sqrt(2*n+1) + 1/2) = round(sqrt(2*n+1)). - Ridouane Oudra, Dec 01 2019
a(n) = ceiling((-1 + sqrt(9 + 8*n))/2) * (1 - ((1/2)*ceiling((1 + sqrt(9 + 8*n))/2))) + n. - Ryan Jean, Sep 03 2022
G.f.: x*y/((1 - x)*(1 - x*y)^2). - Stefano Spezia, Feb 21 2024

New name from Omar E. Pol, Jul 15 2012

Typo in definition fixed by Reinhard Zumkeller, Aug 05 2015

A048760 Largest square <= n.

Original entry on oeis.org

0, 1, 1, 1, 4, 4, 4, 4, 4, 9, 9, 9, 9, 9, 9, 9, 16, 16, 16, 16, 16, 16, 16, 16, 16, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 36, 36, 36, 36, 36, 36, 36, 36, 36, 36, 36, 36, 36, 49, 49, 49, 49, 49, 49, 49, 49, 49, 49, 49, 49, 49, 49, 49, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64

Offset: 0

Charles T. Le (charlestle(AT)yahoo.com)

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
Krassimir T. Atanassov, On Some of Smarandache's Problems, 1999.
Henry Bottomley, Illustration of A000196, A048760, A053186.
Jose Castillo, Other Smarandache Type Functions: Inferior/Superior Smarandache f-part of x, Smarandache Notions Journal, Vol. 10, No. 1-2-3 (1999), pp. 202-204.
Valentina V. Radeva and Krassimir T. Atanassov, On the 40-th and 41-st Smarandache's problems, Notes on Number Theory and Discrete Mathematics, Vol. 4, No. 3 (1998), pp. 101-104.
Florentin Smarandache, Only Problems, Not Solutions!, 1993.
Michael Somos, Sequences used for indexing triangular or square arrays.

Cf. A000196, A000290, A048761, A345202.

a048760 = (^ 2) . a000196  -- Reinhard Zumkeller, Feb 12 2012

A048760 := proc(n)
    floor(sqrt(n)) ;
    %^2 ;
end proc: # R. J. Mathar, May 19 2016

Array[Floor[Sqrt[#]]^2&,80,0] (* Harvey P. Dale, Mar 30 2012 *)
Table[PadRight[{},2n+1,n^2],{n,0,10}]//Flatten (* Harvey P. Dale, Feb 28 2025 *)

a(n) = sqrtint(n)^2; \\ Michel Marcus, Jun 06 2015

a(n) = floor(n^(1/2))^2 = A000290(A000196(n)). - Reinhard Zumkeller, Feb 12 2012, Sep 03 2002
n^2 repeated (2n+1) times, n=0,1,... - Zak Seidov, Oct 25 2008
Sum_{n>=1} (1/a(n) - 1/n) = gamma + zeta(2) (= A345202). - Amiram Eldar, Jun 12 2021
Sum_{n>=1} 1/a(n)^2 = 2*zeta(3) + Pi^4/90. - Amiram Eldar, Aug 15 2022

A053610 Number of positive squares needed to sum to n using the greedy algorithm.

Original entry on oeis.org

1, 2, 3, 1, 2, 3, 4, 2, 1, 2, 3, 4, 2, 3, 4, 1, 2, 3, 4, 2, 3, 4, 5, 3, 1, 2, 3, 4, 2, 3, 4, 5, 3, 2, 3, 1, 2, 3, 4, 2, 3, 4, 5, 3, 2, 3, 4, 5, 1, 2, 3, 4, 2, 3, 4, 5, 3, 2, 3, 4, 5, 3, 4, 1, 2, 3, 4, 2, 3, 4, 5, 3, 2, 3, 4, 5, 3, 4, 5, 2, 1, 2, 3, 4, 2, 3, 4, 5, 3, 2, 3, 4, 5, 3, 4, 5, 2, 3, 4, 1, 2, 3, 4

Offset: 1

Jud McCranie, Mar 19 2000

nonn

Define f(n) = n - x^2 where (x+1)^2 > n >= x^2. a(n) = number of iterations in f(...f(f(n))...) to reach 0.
a(n) = 1 iff n is a perfect square.
Also sum of digits when writing n in base where place values are squares, cf. A007961. - Reinhard Zumkeller, May 08 2011
The sequence could have started with a(0)=0. - Thomas Ordowski, Jul 12 2014
The sequence is not bounded, see A006892. - Thomas Ordowski, Jul 13 2014

			7=4+1+1+1, so 7 requires 4 squares using the greedy algorithm, so a(7)=4.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A006892 (positions of records), A055401, A007961.

Cf. A000196, A000290, A057945 (summing triangular numbers).

a053610 n = s n $ reverse $ takeWhile (<= n) $ tail a000290_list where
  s _ []                 = 0
  s m (x:xs) | x > m     = s m xs
             | otherwise = m' + s r xs where (m',r) = divMod m x
-- Reinhard Zumkeller, May 08 2011

A053610 := proc(n)
    local a,x;
    a := 0 ;
    x := n ;
    while x > 0 do
        x := x-A048760(x) ;
        a := a+1 ;
    end do:
    a ;
end proc: # R. J. Mathar, May 13 2016

f[n_] := (n - Floor[Sqrt[n]]^2); g[n_] := (m = n; c = 1; While[a = f[m]; a != 0, c++; m = a]; c); Table[ g[n], {n, 1, 105}]

A053610(n,c=1)=while(n-=sqrtint(n)^2,c++);c \\ M. F. Hasler, Dec 04 2008

from math import isqrt
def A053610(n):
    c = 0
    while n:
        n -= isqrt(n)**2
        c += 1
    return c # Chai Wah Wu, Aug 01 2023

a(n) = A007953(A007961(n)). - Henry Bottomley, Jun 01 2000
a(n) = a(n - floor(sqrt(n))^2) + 1 = a(A053186(n)) + 1 [with a(0) = 0]. - Henry Bottomley, May 16 2000
A053610 = A002828 + A062535. - M. F. Hasler, Dec 04 2008

A068527 Difference between smallest square >= n and n.

Original entry on oeis.org

0, 0, 2, 1, 0, 4, 3, 2, 1, 0, 6, 5, 4, 3, 2, 1, 0, 8, 7, 6, 5, 4, 3, 2, 1, 0, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9

Offset: 0

Vladeta Jovovic, Mar 21 2002

The greedy inverse (sequence of the smallest k such that a(k)=n) starts 0, 3, 2, 6, 5, 11, 10, 18, 17, 27, 26, 38, 37, 51, 50, ... and appears to be given by A010000 and A002522, interleaved. - R. J. Mathar, Nov 17 2014

Ivan Panchenko, Table of n, a(n) for n = 0..1000

Cf. A053186, A068869, A066857.

Bisections: A348596, A350962.

[ Ceiling(Sqrt(n))^2-n : n in [0..50] ]; // Wesley Ivan Hurt, Jun 11 2014

A068527:=n->ceil(sqrt(n))^2-n; seq(A068527(n), n=0..100); # Wesley Ivan Hurt, Jun 11 2014

Table[Ceiling[Sqrt[n]]^2-n,{n,0,100}] (* Vladimir Joseph Stephan Orlovsky, Feb 01 2012 *)

a(n)=if(issquare(n), 0, (sqrtint(n)+1)^2-n) \\ Charles R Greathouse IV, Oct 22 2014

from math import isqrt
def A068527(n): return 0 if n == 0 else (isqrt(n-1)+1)**2-n # Chai Wah Wu, Feb 22 2022

a(n) = A048761(n) - n = ceiling(sqrt(n))^2 - n.

G.f.: (-x^2 + (x-x^2)*Sum_{m>=1} (1+2*m)*x^(m^2))/(1-x)^2. This sum is related to Jacobi Theta functions. - Robert Israel, Nov 17 2014

A077116 n^3 - A065733(n).

Original entry on oeis.org

0, 0, 4, 2, 0, 4, 20, 19, 28, 0, 39, 35, 47, 81, 40, 11, 0, 13, 56, 135, 79, 45, 39, 67, 135, 0, 152, 83, 48, 53, 104, 207, 7, 216, 100, 26, 0, 28, 116, 270, 496, 277, 104, 546, 503, 524, 615, 139, 368, 0, 391, 155, 732, 652, 648, 726, 55, 293, 631, 170, 704

Offset: 0

Reinhard Zumkeller, Oct 29 2002

a(n) = 0 for n = m^2. - Zak Seidov, May 11 2007

It has been asked whether some primes do not occur in this sequence. It seems indeed that primes 3, 5, 17, 23, 29, 31, 37, 41, 43, 59, 61,... do not occur, primes 2, 7, 11, 13, 19, 47, 53, 67, 79, 83,... do. For further investigations, see A087285 = the range of this sequence, and also the related sequences A229618 = range of A181138, and A165288. - M. F. Hasler, Sep 26 2013 and Oct 05 2013

			A065733(10) = 961 = 31^2 is the largest square less than or equal to 10^3 = 1000, therefore a(10) = 1000 - 961 = 39.

Cf. A000578, A070929, A077118, A077119, A075847.

A077116 := proc(n)
    A053186(n^3) ;
end proc: # R. J. Mathar, Jul 12 2016

Table[c = n^3; c - Floor[Sqrt[c]]^2, {n, 0, 200}] (* Vladimir Joseph Stephan Orlovsky, Jun 02 2011 *)

A077116(n)=n^3-sqrtint(n^3)^2 \\ - M. F. Hasler, Sep 26 2013

a(n) = A154333(n) unless n is a square or, equivalently, a(n)=0. - M. F. Hasler, Oct 05 2013

a(n) = A053186(n^3). - R. J. Mathar, Jul 12 2016

A071797 Restart counting after each new odd integer (a fractal sequence).

Original entry on oeis.org

1, 1, 2, 3, 1, 2, 3, 4, 5, 1, 2, 3, 4, 5, 6, 7, 1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11

Offset: 1

Antonio Esposito, Jun 06 2002

The following sequences all have the same parity: A004737, A006590, A027052, A071028, A071797, A078358, A078446.
This is also a triangle read by rows in which row n lists the first 2*n-1 positive integers, n >= 1 (see example). - Omar E. Pol, May 29 2012
a(n) mod 2 = A071028(n). - Boris Putievskiy, Jul 24 2013
The triangle in the example is the triangle used by Kircheri in 1664. See the link "Mundus Subterraneus". - Charles Kusniec, Sep 11 2022

			a(1)=1; a(9)=5; a(10)=1;
From _Omar E. Pol_, May 29 2012: (Start)
Written as a triangle the sequence begins:
  1;
  1, 2, 3;
  1, 2, 3, 4, 5;
  1, 2, 3, 4, 5, 6, 7;
  1, 2, 3, 4, 5, 6, 7, 8, 9;
  1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11;
  1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13;
  1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15;
Row n has length 2*n - 1 = A005408(n-1). (End)

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000
Glen Joyce C. Dulatre, Jamilah V. Alarcon, Vhenedict M. Florida, Daisy Ann A. Disu, On Fractal Sequences, DMMMSU-CAS Science Monitor (2016-2017) Vol. 15 No. 2, 109-113.
C. Kimberling, Numeration systems and fractal sequences, Acta Arithmetica 73 (1995) 103-117.
Athanasii Kircheri, Mundus Subterraneus, (1664), pg. 24.
F. Smarandache, Only Problems, Not Solutions!, Phoenix,AZ: Xiquan,1993.
Michael Somos, Sequences used for indexing triangular or square arrays

Cf. A002260, A004737, A006590, A027052, A071028, A078358, A078446.
Cf. A074294.
Row sums give positive terms of A000384.

import Data.List (inits)
a071797 n = a071797_list !! (n-1)
a071797_list = f $ tail $ inits [1..] where
   f (xs:_:xss) = xs ++ f xss
-- Reinhard Zumkeller, Apr 14 2014

A071797 := proc(n)
    n-A048760(n-1) ;
end proc: # R. J. Mathar, May 29 2016

Array[Range[2# - 1]&, 10] // Flatten (* Jean-François Alcover, Jan 30 2018 *)

PARI
```
a(n)=if(n<1,0,n-sqrtint(n-1)^2)
```

a(n) = 1 + A053186(n-1).
a(n) = n - 1 - ceiling(sqrt(n))*(ceiling(sqrt(n))-2); n > 0.
a(n) = n - floor(sqrt(n-1))^2, distance between n and the next smaller square. - Marc LeBrun, Jan 14 2004

cp's OEIS Frontend

A176864 Numbers k such that A053186(k) is prime.

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A010052 Characteristic function of squares: a(n) = 1 if n is a square, otherwise 0.

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A000196 Integer part of square root of n. Or, number of positive squares <= n. Or, n appears 2n+1 times.

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A005563 a(n) = n*(n+2) = (n+1)^2 - 1.

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A002262 Triangle read by rows: T(n,k) = k, 0 <= k <= n, in which row n lists the first n+1 nonnegative integers.

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