A001044 -id:A001044 - cp's OEIS frontend

A065886 Smallest square divisible by n!.

1, 1, 4, 36, 144, 3600, 3600, 176400, 2822400, 25401600, 25401600, 3073593600, 110649369600, 18699743462400, 74798973849600, 1869974346240000, 29919589539840000, 8646761377013760000, 77820852393123840000, 28093327713917706240000, 112373310855670824960000

Offset: 0

Henry Bottomley, Nov 27 2001

nonn

			a(10) = 25401600 since 10! = 3628800 and the smallest square divisible by this is 25401600 = 3628800*7 = 5040^2

Robert Israel, Table of n, a(n) for n = 0..407

N:= 50: # to get a(0)..a(N)
P:= select(isprime, [$2..N]):
nP:= nops(P):
V:= Vector(nP):
A[0]:= 1:
for n from 1 to N do
  for i from 1 to nP do V[i]:= V[i] + padic:-ordp(n,P[i]) od;
  A[n]:= mul(P[i]^(2*ceil(V[i]/2)),i=1..nP)
od:
seq(A[n],n=0..N); # Robert Israel, Jan 30 2017

ssd[n_]:=Module[{nf=n!,k=1},While[!IntegerQ[Sqrt[k*nf]],k++];k*nf]; Array[ssd,20,0] (* Harvey P. Dale, Apr 29 2012 *)

a(n) = A053143(A000142(n)) = A065887(n)^2 = A000142(n)*A055204(n) = A001044(n)/A055071(n)

A067656 Numbers n such that n!*B(2n) is an integer, where B(2n) are the Bernoulli numbers.

Original entry on oeis.org

7, 13, 17, 19, 24, 25, 27, 31, 32, 34, 37, 38, 43, 45, 47, 49, 55, 57, 59, 61, 62, 64, 67, 71, 73, 76, 77, 79, 80, 84, 85, 87, 91, 92, 93, 94, 97, 101, 103, 104, 107, 109, 110, 115, 117, 118, 121, 122, 123, 124, 127, 129, 132, 133, 137, 139, 142, 143, 144, 145, 147

Offset: 1

Benoit Cloitre, Feb 03 2002

nonn

A045979(n), Bernoulli numbers with denominators 6, are included in the sequence.

Also numbers n such that both n+1 and 2n+1 are not prime. - Alexander Adamchuk, Oct 05 2006

Alexander Adamchuk, Oct 05 2006, Table of n, a(n) for n = 1..565

Cf. A000330, A001044.

Cf. A166602. - R. J. Mathar, Feb 14 2010

Select[Range[2,1000],Numerator[ #(#+1)(2#+1)/6/#!^2]==1&] (* Alexander Adamchuk, Oct 05 2006 *)
Select[Range[1000],!PrimeQ[ #+1]&&!PrimeQ[2#+1]&] (* Alexander Adamchuk, Oct 05 2006 *)

Also numbers n>1 such that A000330[n] = Sum[k^2,{k,1,n}] = n(n+1)(2n+1)/6 divides A001044[n] = Product[k^2,{k,1,n}] = (n!)^2. Also numbers n>1 such that Numerator[n(n+1)(2n+1)/6 /(n!)^2] = 1. - Alexander Adamchuk, Oct 05 2006

A162993 The second left hand column of triangle A162990.

Original entry on oeis.org

9, 144, 3600, 129600, 6350400, 406425600, 32920473600, 3292047360000, 398337730560000, 57360633200640000, 9693947010908160000, 1900013614137999360000, 427503063181049856000000, 109440784174348763136000000, 31628386626386792546304000000

Offset: 2

Johannes W. Meijer, Jul 21 2009

A001044 and A162994 are two other left hand columns.

A001710(n+1) equals the square root of a(n).

Array[((#+1)!/2)^2 &, 20, 2] (* Paolo Xausa, Apr 01 2024 *)

a(n) = ((n+1)!/2)^2 for n = 2, 3, ... .

a(n) = A001710(n+1)^2.

A371898 Triangle read by rows: T(n, k) = n * k * (T(n-1, k-1) + T(n-1, k)) for k > 0 with initial values T(n, 0) = 1 and T(i, j) = 0 for j > i.

Original entry on oeis.org

1, 1, 1, 1, 4, 4, 1, 15, 48, 36, 1, 64, 504, 1008, 576, 1, 325, 5680, 22680, 31680, 14400, 1, 1956, 72060, 510480, 1304640, 1382400, 518400, 1, 13699, 1036224, 12233340, 50823360, 94046400, 79833600, 25401600, 1, 109600, 16798768, 318469536, 2017814400, 5794790400, 8346240000, 5893171200, 1625702400

Offset: 0

Werner Schulte, Apr 11 2024

			Lower triangular array starts:
n\k :  0      1        2         3         4         5         6         7
==========================================================================
  0 :  1
  1 :  1      1
  2 :  1      4        4
  3 :  1     15       48        36
  4 :  1     64      504      1008       576
  5 :  1    325     5680     22680     31680     14400
  6 :  1   1956    72060    510480   1304640   1382400    518400
  7 :  1  13699  1036224  12233340  50823360  94046400  79833600  25401600
  etc.

Cf. A000012 (column 0), A007526 (column 1), A001044 (main diagonal).

Cf. A000166, A048993, A131689, A320031, A371766, A371767.

T[n_, k_] := Sum[(-1)^(k - j)*Binomial[k, j]*HypergeometricPFQ[{1, -n}, {}, -j], {j, 0, k}];
Table[T[n, k], {n, 0, 8}, {k, 0, n}] // Flatten  (* Peter Luschny, Apr 12 2024 *)

T(n, k) = if(k==0, 1, if(k > n, 0, n*k*(T(n-1, k-1) + T(n-1, k))))

T(n, k) = Sum_{i=k..n} A131689(i, k) * n! / (n-i)!.
T(n, k) = n! * k! * (Sum_{i=0..n-k} A048993(n-i, k) / i!).
T(n, k) = Sum_{i=0..k} (-1)^(k-i) * binomial(k, i) * A320031(n, i).
Conjecture: E.g.f. of column k is exp(t) * t^k * k! / (Prod_{i=0..k} (1 - i*t)).
Conjecture: Sum_{k=0..n} (-1)^(n-k) * T(n, k) = A000166(n).
T(n, k) = A371766(n, k) * A371767(n, k). - Peter Luschny, Apr 14 2024

A384043 a(n) = [x^n] Product_{k=1..n} (1 + k^2x) / (1 - k^2x).

Original entry on oeis.org

1, 2, 50, 4188, 735600, 221302710, 101667388082, 66218673102680, 58048466179356672, 65901249246347377770, 94061755750395244537250, 164863945136411230998746612, 348110204753572939058548570000, 871547135491620353615820806025918, 2552918049709989779004770502542335650

Offset: 0

Vaclav Kotesovec, May 18 2025

nonn

Cf. A001044, A298851.

Cf. A350366, A384044, A351764.

Table[SeriesCoefficient[Product[(1+k^2*x)/(1-k^2*x), {k, 1, n}], {x, 0, n}], {n, 0, 15}]

a(n) ~ c * d^n * n!^2 / n^(3/2), where d = 16.6871576653578743696262746377576281620174969944584774545888... and c = 0.1371163625236187865398447973928851799479072107076663329994...

A046032 a(n) = (n!)^2 - 1.

Original entry on oeis.org

0, 3, 35, 575, 14399, 518399, 25401599, 1625702399, 131681894399, 13168189439999, 1593350922239999, 229442532802559999, 38775788043632639999, 7600054456551997439999, 1710012252724199423999999, 437763136697395052543999999, 126513546505547170185215999999

Offset: 1

Eric W. Weisstein

Vincenzo Librandi, Table of n, a(n) for n = 1..100
Eric Weisstein's World of Mathematics, Factorial

Cf. A001044.

[Factorial(n)^2-1: n in [1..15]]; // Vincenzo Librandi, Jun 03 2013

Table[(n!)^2-1,{n,30}] (* Vladimir Joseph Stephan Orlovsky, May 19 2011 *)

a(n)=n!^2-1 \\ Charles R Greathouse IV, Sep 19 2012

A060080 Scaled sums of squares.

Original entry on oeis.org

1, 5, 56, 1080, 31680, 1310400, 72576000, 5181926400, 463325184000, 50697529344000, 6663103856640000, 1035678099456000000, 187913434365296640000, 39357424864287129600000, 9424067526124476825600000, 2558178330075402338304000000, 781407199004850168791040000000

Offset: 1

Wolfdieter Lang, Mar 16 2001

nonn

Harry J. Smith, Table of n, a(n) for n = 1..201 [offset shifted by _Georg Fischer_, May 11 2021]

See Comments section at A060074.

Cf. A000330, A001044.

[Factorial(n+1) * Factorial(n-1) * (2*n+1)/6: n in [1..20]]; // Vincenzo Librandi, Jul 05 2018

Table[(n + 1)! (n - 1)! (2 n + 1) / 3!, {n, 1, 30}] (* Vincenzo Librandi Jul 04 2018 *)

a(n) = { (n + 1)!*(n - 1)!*(2*n + 1)/6 } \\ Harry J. Smith, Jul 01 2009

a(n) = (Sum_{k=1..n} k^2)*((n-1)!)^2 = (n+1)!*(n-1)!*(2*n+1)/3!.
a(n) = A000330(n) * A001044(n-1).
From Amiram Eldar, May 03 2025: (Start)
Sum_{n>=1} 1/a(n) = 3 * BesselI(1, 2) (Pi * StruveL(0, 2) + 2) - 3 * BesselI(0, 2) * (Pi * StruveL(1, 2) + 2).
Sum_{n>=1} (-1)^(n+1)/a(n) = 3 * BesselJ(1, 2) * (Pi * StruveH(0, 2) - 2) - 3 * BesselJ(0, 2) * (Pi * StruveH(1, 2) - 2). (End)

Offset changed from 0 to 1 by Georg Fischer, May 09 2021

A092096 a(n) = Sum_{i=0,1,2,..; n-ki >= -n} |n-ki| for k=5.

Original entry on oeis.org

11, 12, 20, 20, 30, 31, 32, 45, 45, 60, 61, 62, 80, 80, 100, 101, 102, 125, 125, 150, 151, 152, 180, 180, 210, 211, 212, 245, 245, 280, 281, 282, 320, 320, 360, 361, 362, 405, 405, 450, 451, 452, 500, 500, 550, 551, 552, 605, 605, 660, 661, 662, 720, 720, 780

Offset: 6

Jahan Tuten (jahant(AT)indiainfo.com), Mar 29 2004

nonn

F. Smarandache, Back and Forth Factorials, Arizona State Univ., Special Collections, 1972.
F. Smarandache, Back and Forth Summants, Arizona State Univ., Special Collections, 1972.

J. Dezert, ed., Smarandacheials (1), Mathematics Magazine for Grades 1-12, No. 4, 2004.
J. Dezert, ed., Smarandacheials (2), Mathematics Magazine for Grades 1-12, No. 4, 2004.
F. Smarandache, Summants [Broken link]

Cf. A001044, A092396, A092397, A092398, A092399, A092971, A092972, A092973, A092974.

S := proc(n,k) local a,i ; a :=0 ; i := 0 ; while n-k*i >= -n do a := a+abs(n-k*i) ; i := i+1 ; od: RETURN(a) ; end: k := 5: seq(S(n,k),n=k+1..80) ; # R. J. Mathar, Feb 01 2008

a[n_] := Sum[Abs[n-5i], {i, 0, Quotient[2n, 5]}];
Table[a[n], {n, 6, 60}] (* Jean-François Alcover, Apr 29 2023 *)

Empirical g.f.: -x^6*(10*x^10-5*x^9-3*x^7-x^6-21*x^5+10*x^4+8*x^2+x+11) / ((x-1)^3*(x^4+x^3+x^2+x+1)^2). - Colin Barker, Jul 28 2013

Edited and extended by R. J. Mathar, Feb 01 2008

Revised by N. J. A. Sloane, Jul 03 2017

A105350 Largest squared factorial dividing n!.

Original entry on oeis.org

1, 1, 1, 1, 4, 4, 36, 36, 576, 576, 518400, 518400, 518400, 518400, 25401600, 25401600, 1625702400, 1625702400, 131681894400, 131681894400, 13168189440000, 13168189440000, 1593350922240000, 1593350922240000, 229442532802560000, 229442532802560000

Offset: 0

Reinhard Zumkeller, Apr 01 2005

nonn

a(n) = A001044(A056039(n)) = A056038(n)^2.

Whenever n > 1 is not in A056067, a(n) = A180064(n). - Andrey Zabolotskiy, Oct 19 2023

Index entries for sequences related to factorial numbers.

Cf. A000142, A000290, A055071, A055772, A056067.

Cf. A180064.

a[n_] := (For[k = 1, Divisible[n!, k!^2], k++]; (k-1)!^2)
Table[a[n], {n, 0, 24}] (* Jean-François Alcover, Aug 07 2018 *)

Data and offset corrected by Jean-François Alcover, Aug 07 2018

Edited by Andrey Zabolotskiy, Oct 18 2023

A133401 Diagonal of polygorial array T(n,k) = n-th polygorial for k = n, for n > 2.

Original entry on oeis.org

18, 576, 46200, 7484400, 2137544640, 981562982400, 678245967907200, 670873729125600000, 913601739437346960000, 1660189302321994373529600, 3923769742187622047360640000, 11805614186177306251101945600000, 44403795869109177300313209696000000

Offset: 3

Jonathan Vos Post, Nov 25 2007

Array T(n,k) = k-th polygorial(n,k) begins:
k  |  polygorial(n,k)
| 1 1  3  18   180    2700     56700     1587600      57153600
| 1 1  4  36   576   14400    518400    25401600    1625702400
| 1 1  5  60  1320   46200   2356200   164934000   15173928000
| 1 1  6  90  2520  113400   7484400   681080400   81729648000
| 1 1  7 126  4284  235620  19085220  2137544640  316356606720
| 1 1  8 168  6720  436800  41932800  5577062400  981562982400
| 1 1  9 216  9936  745200  82717200 12738448800 2598643555200
| 1 1 10 270 14040 1193400 150368400 26314470000 6104957040000

			a(3) = polygorial(3,3) = A006472(3) = product of the first 3 triangular numbers = 1*3*6 = 18.
a(4) = polygorial(4,4) = A001044(4) = product of the first 4 squares = 1*4*9*16 = 576.
a(5) = polygorial(5,5) = A084939(5) = product of the first 5 pentagonal numbers = 1*5*12*22*35 = 46200.

Nathaniel Johnston, Table of n, a(n) for n = 3..100
Daniel Dockery, Polygorials, Special "Factorials" of Polygonal Numbers, preprint, 2003.

Cf. A006472, A001044, A000680, A084939, A084940, A084941, A084942, A084943, A084944, A085356.

A133401 := proc(n) return mul((n/2-1)*m^2-(n/2-2)*m,m=1..n): end: seq(A133401(n),n=3..15); # Nathaniel Johnston, May 05 2011

Table[Product[m*(4 - n + m*(n-2))/2, {m, 1, n}],{n, 3, 20}] (* Vaclav Kotesovec, Feb 20 2015 *)
Table[FullSimplify[(n-2)^n * Gamma[n+1] * Gamma[n+2/(n-2)] / (2^n*Gamma[2/(n-2)])],{n,3,15}] (* Vaclav Kotesovec, Feb 20 2015 *)
polygorial[k_, n_] := FullSimplify[ n!/2^n (k -2)^n*Pochhammer[2/(k - 2), n]]; Array[ polygorial[#, #] &, 13, 3] (* Robert G. Wilson v, Dec 13 2016 *)

a(n) ~ Pi * n^(3*n-1) / (2^(n-2) * exp(2*n+2)). - Vaclav Kotesovec, Feb 20 2015

Edited by Nathaniel Johnston, May 05 2011

cp's OEIS Frontend

A065886 Smallest square divisible by n!.

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A067656 Numbers n such that n!*B(2n) is an integer, where B(2n) are the Bernoulli numbers.

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A162993 The second left hand column of triangle A162990.

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A371898 Triangle read by rows: T(n, k) = n * k * (T(n-1, k-1) + T(n-1, k)) for k > 0 with initial values T(n, 0) = 1 and T(i, j) = 0 for j > i.

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A384043 a(n) = [x^n] Product_{k=1..n} (1 + k^2*x) / (1 - k^2*x).

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A046032 a(n) = (n!)^2 - 1.

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Magma

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A060080 Scaled sums of squares.

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A092096 a(n) = Sum_{i=0,1,2,..; n-k*i >= -n} |n-k*i| for k=5.

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A384043 a(n) = [x^n] Product_{k=1..n} (1 + k^2x) / (1 - k^2x).

A092096 a(n) = Sum_{i=0,1,2,..; n-ki >= -n} |n-ki| for k=5.