A055228 -id:A055228 - cp's OEIS frontend

A306014 Numbers k such that A055228(k)^2 - A055228(k) is a multiple of k, where A055228(k) is ceiling(sqrt(k!)).

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 14, 16, 28, 29, 30, 42, 46, 50, 52, 99, 134, 148, 165, 205, 245, 249, 315, 390, 441, 461, 525, 560, 763, 962, 1596, 1666, 1716, 1847, 1854, 1860, 3515, 4501, 5179, 6850, 7345, 7867, 8940, 9491, 9523, 15688, 19988, 23574, 23956

Offset: 1

Ivan Stoykov, Jun 17 2018

nonn

			For k=6, A055228(6) = ceiling(sqrt(6!)) = 27, and 27^2-27 = 702, which is a multiple of 6.

Hazewinkel, Michiel, ed. (2001) [1994], Gamma Function, Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4

Chai Wah Wu, Table of n, a(n) for n = 1..89 (all terms up to 10^6, n = 1..70 from Jon E. Schoenfield)

Cf. A000188, A055228.

Select[Range[4600],Divisible[Ceiling[Sqrt[#!]]^2-Ceiling[Sqrt[#!]],#]&] (* Harvey P. Dale, Mar 02 2023 *)

default(realprecision,10^5); for(n=1,10^4, if( Mod( ceil(sqrt(n!)) - ceil(sqrt(n!))^2 , n) == 0, print1(n,", "))); \\ Joerg Arndt, Jun 17 2018

Terms > 99 from Joerg Arndt, Jun 17 2018

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

A214080 a(n) = (floor(sqrt(n)))!

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 2, 2, 2, 6, 6, 6, 6, 6, 6, 6, 24, 24, 24, 24, 24, 24, 24, 24, 24, 120, 120, 120, 120, 120, 120, 120, 120, 120, 120, 120, 720, 720, 720, 720, 720, 720, 720, 720, 720, 720, 720, 720, 720, 5040, 5040, 5040, 5040, 5040, 5040, 5040, 5040, 5040

Offset: 0

Mohammad K. Azarian, Dec 22 2012

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Cf. A214078, A214079, A214080, A214081, A214083, A055228, A055226.

Cf. A000142, A000196.

PROG(y := [], n := 50, LOOP(IF( = -1, RETURN y), y := ADJOIN(FLOOR(SQRT(n))!, y), n := n - 1))

[Factorial(Floor(Sqrt(n))): n in [0..60]]; // Vincenzo Librandi, Feb 13 2013

Table[Floor[Sqrt[n]]!, {n, 0, 100}] (* T. D. Noe, Dec 23 2012 *)

a(n) = floor(sqrt(n))!; \\ Altug Alkan, Jan 11 2016

Sum_{n>=0} 1/a(n) = 3*e. - Amiram Eldar, Aug 15 2022

a(n) = A000142(A000196(n)). - Michel Marcus, Aug 15 2022

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

Original entry on oeis.org

1, 1, 1, 1, 2, 4, 8, 17, 34, 71, 153, 341, 782, 1839, 4434, 10935, 27555, 70852, 185686, 495486, 1344956, 3710632, 10397338, 29568648, 85290741, 249391641, 738821756, 2216465268, 6730493989, 20678209929, 64252006059, 201840008711, 640802084315

Offset: 0

Mohammad K. Azarian, Dec 22 2012

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Cf. A214078, A214079, A214080, A214081, A214083, A055228, A055226.

PROG(y := [], n := 35, LOOP(IF(n = -1, RETURN y), y := ADJOIN(FLOOR(n!^(1/3)), y), n := n - 1))

[Floor(Factorial(n)^(1/3)): n in [0..40]]; // Vincenzo Librandi, Feb 08 2013

Table[Floor[n!^(1/3)], {n, 0, 60}] (* Vincenzo Librandi, Feb 08 2013 *)

a(n) = sqrtnint(n!,3); \\ Michel Marcus, Jan 11 2016

A214078 a(n) = (ceiling (sqrt(n)))!.

Original entry on oeis.org

1, 1, 2, 2, 2, 6, 6, 6, 6, 6, 24, 24, 24, 24, 24, 24, 24, 120, 120, 120, 120, 120, 120, 120, 120, 120, 720, 720, 720, 720, 720, 720, 720, 720, 720, 720, 720, 5040, 5040, 5040, 5040, 5040, 5040, 5040, 5040, 5040, 5040, 5040, 5040, 5040, 40320, 40320, 40320

Offset: 0

Mohammad K. Azarian, Dec 22 2012

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Cf. A214078, A214079, A214080, A214081, A214083, A055228, A055226.

Cf. A000142, A003059.

PROG(y := [], n := 50, LOOP(IF(n = -1, RETURN y), y := ADJOIN(CEILING(SQRT(n))!, y), n := n - 1))

[Factorial(Ceiling (Sqrt(n))): n in [0..50]]; // Vincenzo Librandi, Feb 13 2013

Table[Ceiling[Sqrt[n]]!, {n, 0, 50}] (* T. D. Noe, Dec 23 2012 *)

a(n) = ceil(sqrt(n))!; \\ Altug Alkan, Jan 11 2016

from math import factorial, isqrt
def A214078(n): return factorial(1+isqrt(n-1)) if n else 1 # Chai Wah Wu, Jul 28 2022

a(n) = A000142(A003059(n)). - Michel Marcus, Jul 28 2022

Sum_{n>=0} 1/a(n) = e + 2. - Amiram Eldar, Aug 15 2022

A214079 a(n) = ceiling( n^(1/3) )!.

Original entry on oeis.org

1, 1, 2, 2, 2, 2, 2, 2, 2, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 120, 120, 120, 120, 120

Offset: 0

Mohammad K. Azarian, Dec 22 2012

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Cf. A214078, A214079, A214080, A214081, A214083, A055228, A055226.

Cf. A000142, A134914.

PROG(y := [], x := 70, LOOP(IF(x = -1, RETURN y), y := ADJOIN(CEILING(x^(1/3))!, y), x := x - 1))

[Factorial(Ceiling(n^(1/3))): n in [0..80]]; // Vincenzo Librandi, Feb 13 2013

Table[Ceiling[n^(1/3)]!, {n, 0, 100}] (* T. D. Noe, Dec 23 2012 *)
Ceiling[Surd[Range[0,70],3]]! (* Harvey P. Dale, Nov 19 2022 *)

a(n) = ceil(n^(1/3))! \\ Altug Alkan, Jan 11 2016

Sum_{n>=0} 1/a(n) = 4*e. - Amiram Eldar, Aug 15 2022

a(n) = A000142(A134914(n)). - Michel Marcus, Aug 15 2022

A214081 a(n) = floor( n^(1/3) )!.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24

Offset: 0

Mohammad K. Azarian, Dec 22 2012

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Cf. A055226, A055228, A214078, A214079, A214080, A214081, A214083.

Cf. A000142, A048766.

PROG(y := [], n := 70, LOOP(IF(n = -1, RETURN y), y := ADJOIN(FLOOR(n^(1/3))!, y), n := n - 1))

[Factorial(Floor(n^(1/3))): n in [0..80]]; // Vincenzo Librandi, Feb 13 2013

Table[Floor[n^(1/3)]!, {n, 0, 100}] (* T. D. Noe, Dec 23 2012 *)
Floor[CubeRoot[Range[0,90]]]! (* Harvey P. Dale, Jan 15 2024 *)

a(n) = floor(n^(1/3))!; \\ Altug Alkan, Jan 11 2016

Sum_{n>=0} 1/a(n) = 10*e. - Amiram Eldar, Aug 15 2022

a(n) = A000142(A048766(n)). - Michel Marcus, Aug 15 2022

A214049 Least m>0 such that n! <= m^3.

Original entry on oeis.org

1, 2, 2, 3, 5, 9, 18, 35, 72, 154, 342, 783, 1840, 4435, 10936, 27556, 70853, 185687, 495487, 1344957, 3710633, 10397339, 29568649, 85290742, 249391642, 738821757, 2216465269, 6730493990, 20678209930, 64252006060, 201840008712, 640802084316, 2055394684174

Offset: 1

Clark Kimberling, Jul 18 2012

			a(4) = 3 because 2^3 < 4! <= 3^3.

Clark Kimberling, Table of n, a(n) for n = 1..1000

Cf. A055228.

Mathematica
```
Table[Ceiling[n!^(1/3)], {n,1,40}]
```

A308852 Minimum number k such that the k-th tetrahedral number is not smaller than n!.

Original entry on oeis.org

1, 2, 3, 5, 8, 16, 31, 62, 129, 279, 621, 1421, 3343, 8057, 19870, 50071, 128747, 337414, 900358, 2443947, 6742667, 18893218, 53729800, 154983562, 453174686, 1342528227, 4027584682, 12230119228, 37574801086, 116753643340, 366767636286, 1164414663338, 3734900007009

Offset: 1

Lorenzo Sauras Altuzarra, Jun 28 2019

nonn

More formally, a(n) is the minimum element of the set of positive integers k such that the k-th tetrahedral number is not smaller than the n-th factorial.

Open problem: what is the cardinality of the set of numbers that are simultaneously a tetrahedral number and a factorial number? For example, 1 and 120 belong to this set.

			The minimum tetrahedral number not smaller than 4! is 35 (i.e., the 5th tetrahedral number), so a(4) = 5.
The minimum tetrahedral number not smaller (equal, in fact) than 5! is 120 (i.e., the 8th tetrahedral number), so a(5) = 8.

Cf. A000142 (factorial numbers), A000292 (tetrahedral numbers).

Cf. A055228 (for n^2), A214049 (for n^3), A214448 (for n^4).

Floor[(6 Range[33]!)^(1/3)] (* Giovanni Resta, Jul 30 2019 *)

a(n) = {my(k=1); while (k*(k+1)*(k+2)/6 < n!, k++); k;} \\ Michel Marcus, Jun 28 2019

a(n) = ceiling((sqrt(3) * sqrt(243*(n!)^2 - 1) + 27*n!)^(1/3) / 3^(2/3) + 1/(3^(1/3) * (sqrt(3) * sqrt(243*(n!)^2 - 1) + 27*n!)^(1/3)) - 1). - Daniel Suteu, Jun 30 2019

a(n) = floor((6*n!)^(1/3)). - Giovanni Resta, Jul 30 2019

a(26)-a(33) from Daniel Suteu, Jun 30 2019

A074188 Smallest power (>=2) >= n!.

Original entry on oeis.org

1, 4, 8, 25, 121, 729, 5041, 40401, 363609, 3629025, 39917124, 479040769, 6227103744, 87178467600, 1307674583296, 20922793332736, 355687454263684, 6402373820077225, 121645100663836929, 2432902009335560361

Offset: 1

Amarnath Murthy, Aug 31 2002

nonn

			a(3) = 8 = 2^3 > 6=3!.

Cf. A074187.

Cf. A055228. It seems that a(n)=A055228(n)^2 for n>3.

More terms from Hugo Pfoertner, Jul 20 2004

cp's OEIS Frontend

A306014 Numbers k such that A055228(k)^2 - A055228(k) is a multiple of k, where A055228(k) is ceiling(sqrt(k!)).

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

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A214080 a(n) = (floor(sqrt(n)))!

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

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A214078 a(n) = (ceiling (sqrt(n)))!.

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A214079 a(n) = ceiling( n^(1/3) )!.

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A214081 a(n) = floor( n^(1/3) )!.

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