A066857 -id:A066857 - cp's OEIS frontend

A068527 Difference between smallest square >= n and n.

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

A068869 Smallest number k such that n! + k is a square.

Original entry on oeis.org

0, 2, 3, 1, 1, 9, 1, 81, 729, 225, 324, 39169, 82944, 176400, 215296, 3444736, 26167684, 114349225, 255004929, 1158920361, 11638526761, 42128246889, 191052974116, 97216010329, 2430400258225, 1553580508516, 4666092737476, 565986718738441, 2137864362693921

Offset: 1

Amarnath Murthy, Mar 13 2002

nonn

Observation: for n < 2000, only for n = 1, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 15, 16 is a(n) a square (see A360210).
According to my conjecture that n! + n^2 != m^2 for n >= 1, m >= 0 (see A004664), for all terms: a(n) != n^2. - Alexander R. Povolotsky, Oct 06 2008
There are two cases: a(n) > sqrt(n!) in A182203 and a(n) < sqrt(n!) in A182204. - Artur Jasinski, Apr 13 2012

			a(6) = 9 as 6! + 9 = 729 is a square.

T. D. Noe, Table of n, a(n) for n=1..100

Cf. A038202, A048761, A055228, A066857, A360210.

Cf. A182203, A182204.

Table[ Ceiling[ Sqrt[n! ]]^2 - n!, {n, 1, 28}]

A068869(n)=(sqrtint(n!-1)+1)^2-n!  \\ M. F. Hasler, Apr 01 2012

from math import factorial, isqrt
def a(n): return (isqrt((f:=factorial(n))-1)+1)**2 - f
print([a(n) for n in range(1, 30)]) # Michael S. Branicky, Jan 30 2023

a(n) = A055228(n)^2 - n! = ceiling(sqrt(n!))^2 - n! = A048761(n!) - n!.

a(n) <= A038202(n)^2, with equality for the n listed in the first comment. - M. F. Hasler, Apr 01 2012

More terms from Vladeta Jovovic, Mar 21 2002

Edited by Robert G. Wilson v and N. J. A. Sloane, Mar 22 2002

A226973 Difference between n! and the largest cube < n!.

Original entry on oeis.org

1, 1, 5, 16, 56, 208, 127, 1016, 4969, 47223, 264979, 789832, 7668081, 4272696, 130217625, 883909125, 9969785792, 52152119144, 128092980744, 2166664965184, 29992267884032, 272465658461528, 1588888484126208, 10747891377020979, 5480400487212279, 70703132766750784, 1908984584702271168

Offset: 1

Zak Seidov, Jun 25 2013

nonn

Also, smallest number k such that n! - k is a cube.

Sequence is not monotonic: a(n) < a(n-1) for n: 7, 14, 25, 30, 51, 106, 168, 279, 288.

			a(2) = 2! - 1^3 = 1, a(3) = 3! - 1^3 = 5, a(4) = 4! - 3^3 = 16.

Charles R Greathouse IV, Table of n, a(n) for n = 1..500

Cf. A000142, A066857, A214083.

Join[{1}, Table[n! - Floor[(n!)^(1/3)]^3, {n, 2, 30}]]

a(n)=my(N=n!);N-sqrtnint(N,3)^3 \\ Charles R Greathouse IV, Jun 25 2013

a(n) = n! - floor (n!^(1/3))^3 = A000142(n) - A214083(n)^3.

A240940 Least number k >= 0 such that n! - k is a perfect power.

Original entry on oeis.org

0, 1, 2, 8, 20, 44, 127, 320, 476, 3584, 12311, 4604, 74879, 414119, 2071775, 5703551, 11551671, 45680444, 442548224, 1960632176, 2657058876, 24923993276, 130518272975, 1478154932316, 5446454455004, 38610655379975, 204033398880671, 538347188396016, 3809155729331900, 27460809907547975, 52607402757814775

Offset: 1

Derek Orr, Aug 04 2014

nonn

Cf. A066857 (n! - k is a square), A226973.

a(n)=for(k=0,n!,s=n!-k;if(ispower(s)||s==1,return(k)))
n=1;while(n<50,print1(a(n),", ");n++)

a(n)=for(k=1, n!, if(2^k>n!, kk=k; break)); if(kk==1, return(0)); L=List([]); for(i=2, kk, listinsert(L, n!-floor(n!^(1/i))^i, 1)); listsort(L); L[1]
vector(40, n, a(n)) \\ faster program

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A068527 Difference between smallest square >= n and n.

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A068869 Smallest number k such that n! + k is a square.

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A226973 Difference between n! and the largest cube < n!.

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A240940 Least number k >= 0 such that n! - k is a perfect power.

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