A068869 -id:A068869 - cp's OEIS frontend

A182203 Numbers n for which A068869(n) > sqrt(n!).

2, 3, 9, 12, 13, 17, 18, 21, 22, 23, 28, 31, 32, 34, 35, 36, 38, 39, 40, 41, 44, 48, 49, 51, 52, 56, 57, 58, 63, 65, 67, 68, 69, 71, 72, 73, 75, 77, 78, 79, 80, 81, 82, 85, 86, 87, 88, 90, 91, 92, 93, 97, 99, 100

Offset: 1

Artur Jasinski, Apr 17 2012

nonn

There are two cases: A068869(n) > sqrt(n!) in this sequence and A068869(n) < sqrt(n!) in A182204.

Cf. A068869, A182204.

aa = {}; Do[k = Ceiling[Sqrt[n!]]^2 - n!; If[k > Sqrt[n!], AppendTo[aa, n]], {n, 1, 100}]; aa

A182204 Numbers n for which A068869(n) < sqrt(n!).

Original entry on oeis.org

1, 4, 5, 6, 7, 8, 10, 11, 14, 15, 16, 19, 20, 24, 25, 26, 27, 29, 30, 33, 37, 42, 43, 45, 46, 47, 50, 53, 54, 55, 59, 60, 61, 62, 64, 66, 70, 74, 76, 83, 84, 89, 94, 95, 96, 98, 102, 103, 104, 107, 109, 111, 113, 114, 117, 118, 122, 123, 125, 127, 128, 129

Offset: 1

Artur Jasinski, Apr 17 2012

nonn

There are two cases: A068869(n) > sqrt(n!) see A182203 and A068869(n) < sqrt(n!) this sequence.

Cf. A068869, A182203.

aa = {}; Do[k = Ceiling[Sqrt[n!]]^2 - n!;
  If[k < Sqrt[n!], AppendTo[aa, n]], {n, 1, 118}]; aa

A360210 Indices of squares in A068869.

Original entry on oeis.org

1, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 15, 16

Offset: 1

Alexander R. Povolotsky, Jan 29 2023

a(14) > 30000. - Michel Marcus, Jan 30 2023

			5 is a term because A068869(5) = 1 is a square;
8 is a term because A068869(8) = 81 is a square.

Cf. A068869.

isok(n) = issquare((sqrtint(n!-1)+1)^2-n!); \\ Michel Marcus, Jan 30 2023

A182278 Indices n such that A068869(n)<A068869(n-1).

Original entry on oeis.org

4, 7, 10, 24, 26, 42, 117, 135, 141, 150, 194, 220, 224, 236, 312, 336, 399, 406, 438, 497, 529, 652, 663, 707, 797, 844, 879, 908, 988, 1092, 1099, 1133, 1141, 1300, 1304, 1371, 1397, 1494, 1513, 1536, 1676, 1886, 1970, 1981, 1988, 2076, 2093, 2221, 2270, 2356, 2390, 2462

Offset: 1

M. F. Hasler and Artur Jasinski, Apr 23 2012

nonn

The sequence A068869 is statistically growing, but I conjecture that there are infinitely many indices n such that A068869(n)<A068869(n-1). However, we don't yet know whether there are two consecutive integers in the sequence, i.e., a term n such that, additionally to the above, A068869(n+1)<A068869(n).

a(n) roughly grows like n^2.

w = 0; aa = {}; Do[k = Ceiling[Sqrt[n!]]^2 - n!;
If[k < w, AppendTo[aa, n - 1]]; w = k, {n, 1, 2500}]; aa

for(i=1,2500,A068869(i)<A068869(i-1)&print1(i","))

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

A038202 Least k such that n! + k^2 is a square.

Original entry on oeis.org

1, 1, 3, 1, 9, 27, 15, 18, 288, 288, 420, 464, 1856, 10080, 46848, 210240, 400320, 652848, 3991680, 27528402, 32659200, 163296000, 1143463200, 1305467240, 6840489600, 9453465438, 337082683248, 163425485250, 8376514506360, 8440230839040, 5088099594240

Offset: 4

David W. Wilson

nonn

Let f = n!/4 and let x be the largest divisor of f such that x < sqrt(f). Then a(n) = f/x - x. The greatest k such that n! + k^2 is a square is f-1. The number of k for which n! + k^2 is a square is A038548(n). - T. D. Noe, Nov 02 2004

For greatest k such that n! + k^2 is a square see A181892; for numbers x such that n! + k^2 = x^2 see A181896. - Artur Jasinski, Mar 31 2012

Sudipta Mallick, Table of n, a(n) for n = 4..58
Eric Weisstein's World of Mathematics, Brocard's Problem

Cf. A038548 (number of divisors of n that are at most sqrt(n)), A068869.

Cf. A181892, A181896.

Table[f=n!/4; x=Max[Select[Divisors[f], #<=Sqrt[f]&]]; f/x-x, {n, 4, 20}] (* T. D. Noe, Nov 02 2004 *)

a(n) = my(k=0); while(!issquare(n!+k^2), k++); k; \\ Michel Marcus, Sep 16 2018

a(30)-a(34) from Jon E. Schoenfield, Sep 15 2018

A066857 Smallest number k such that n! - k is a square.

Original entry on oeis.org

0, 1, 2, 8, 20, 44, 140, 320, 476, 3584, 12311, 4604, 74879, 414119, 2071775, 5703551, 11551671, 45680444, 442548224, 1960632176, 2657058876, 24923993276, 130518272975, 1478154932316, 5446454455004, 38610655379975

Offset: 1

Reinhard Zumkeller, Jan 21 2002

nonn

Sequence is not monotonic: a(n) < a(n-1) for n = 12, 71, 90, 143, 145, 151, 172, 218, 257. - Zak Seidov, Jun 25 2013

			a(10) = 3628800 - 1904 * 1904 = 3628800 - 3625216 = 3584.

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

Cf. A068869.

Table[n! - Floor[Sqrt[n! ]]^2, {n, 1, 27}]

a(n)=my(N=n!); N-sqrtint(N)^2 \\ Charles R Greathouse IV, Jun 25 2013

a(n) = A053186(n!) = n!-A048760(n!) = n!-floor(sqrt(n!))^2 = n!-A055226(n)^2.

More terms from Vladeta Jovovic, Mar 21 2002

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

A240937 Least number k >= 0 such that n! + k is a cube.

Original entry on oeis.org

0, 6, 2, 3, 5, 9, 792, 2555, 10368, 23464, 84888, 1047087, 2483200, 54721675, 228537856, 1394007616, 5090444477, 51286309703, 608427634303, 3260058995493, 11314112766137, 51848285189219, 1034026438223449, 11075640379838488, 181108172062981288, 1566869630866485093

Offset: 1

Derek Orr, Aug 03 2014

nonn

Robert Israel, Table of n, a(n) for n = 1..524

Cf. A068869.

f:= proc(n) local N; N:= n!; ceil(N^(1/3))^3 - N end proc:
seq(f(n), n=1..30); # Robert Israel, Aug 04 2014

f[n_] := Block[{c = n! - 1}, (1 + Floor[c^(1/3)])^3 - c - 1]; Array[f, 26] (* Robert G. Wilson v, Aug 04 2014 *)
Table[Ceiling[CubeRoot[n!]]^3-n!,{n,30}] (* Harvey P. Dale, Jun 21 2025 *)

a(n)=for(k=0,10^10,s=n!+k;if((ispower(s)&&ispower(s)%3==0)||s==1,return(k)))
n=1;while(n<20,print1(a(n),", ");n++)

vector(50, n, ceil(n!^(1/3))^3-n!) \\ faster program

a(15) onward from Robert G. Wilson v, Aug 04 2014

A083397 Largest prime factor of n! + k where k is the least positive integer such that n! + k is a square.

Original entry on oeis.org

0, 2, 3, 5, 11, 3, 71, 67, 67, 127, 13, 509, 137, 37, 71471, 71471, 409993, 941351, 24419, 287093, 7147792819, 110647261, 80392811773, 4716679469, 4716679469, 323905128133, 8392290961, 551615338229, 34178276390953, 73669621631

Offset: 1

Jason Earls, Jun 06 2003

nonn

For n > 1, n! cannot be a perfect square. Proof: All exponents of the prime factors of a square are even. But in the factorization of n! at least one of the primes will appear only once due to Bertrand's Postulate which says there is always a prime between m and 2m.

			a(9)=67 because 9!+729 = 363609 = 3^4*67^2 is a square with largest prime factor of 67.

Amiram Eldar, Table of n, a(n) for n = 1..100

Cf. A068869.

Join[{0},Table[FactorInteger[(Floor[Sqrt[n!]]+1)^2][[-1,1]],{n,2,30}]] (* Harvey P. Dale, Jan 04 2012 *)

A240939 Least number k >= 0 such that n! + k is a perfect power.

Original entry on oeis.org

0, 2, 2, 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, 5112360635841936

Offset: 1

Derek Orr, Aug 03 2014

nonn

The only n <= 805 where n! + a(n) is not a square is 3. - Robert Israel, Aug 01 2024

Robert Israel, Table of n, a(n) for n = 1..805

Cf. A068869, A240937.

f:= proc(n) local v,m,p,r;
   m:= infinity;
   v:= n!;
   p:= 1;
   do
     p:= nextprime(p);
     if 2^p >= m+v then break fi;
     r:= ceil(v^(1/p))^p - v;
     if r < m then m:= r fi;
   od;
   m
end proc:
map(f, [$1..50]);

nextPerfectPower[n_] := Min@ Table[(Floor[n^(1/k)] + 1)^k, {k, 2, 1 + Floor@ Log2@ n}]; f[n_] := nextPerfectPower[n!] - n!; f[1] = 0; Array[f, 30] (* Robert G. Wilson v, Aug 04 2014 *)

a(n)=for(k=0,10^10,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,ceil(n!^(1/i))^i-n!,1));listsort(L);L[1]
vector(40, n, a(n)) \\ faster program

a(18) onward from Robert G. Wilson v, Aug 04 2014

cp's OEIS Frontend

A182203 Numbers n for which A068869(n) > sqrt(n!).

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

A182204 Numbers n for which A068869(n) < sqrt(n!).

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

A360210 Indices of squares in A068869.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

PARI

A182278 Indices n such that A068869(n)<A068869(n-1).

Views

Author

Keywords

Comments

Programs

Mathematica

PARI

A068527 Difference between smallest square >= n and n.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Python

Formula

A038202 Least k such that n! + k^2 is a square.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Extensions

A066857 Smallest number k such that n! - k is a square.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A240937 Least number k >= 0 such that n! + k is a cube.

Views

Author

Keywords

Links

Crossrefs

Programs

Maple