A002109 -id:A002109 - cp's OEIS frontend

A343656 Array read by antidiagonals where A(n,k) is the number of divisors of n^k.

1, 1, 1, 1, 2, 1, 1, 3, 2, 1, 1, 4, 3, 3, 1, 1, 5, 4, 5, 2, 1, 1, 6, 5, 7, 3, 4, 1, 1, 7, 6, 9, 4, 9, 2, 1, 1, 8, 7, 11, 5, 16, 3, 4, 1, 1, 9, 8, 13, 6, 25, 4, 7, 3, 1, 1, 10, 9, 15, 7, 36, 5, 10, 5, 4, 1, 1, 11, 10, 17, 8, 49, 6, 13, 7, 9, 2, 1, 1, 12, 11, 19, 9, 64, 7, 16, 9, 16, 3, 6, 1

Offset: 1

Gus Wiseman, Apr 28 2021

First differs from A343658 at A(4,2) = 5, A343658(4,2) = 6.

As a triangle, T(n,k) = number of divisors of k^(n-k).

			Array begins:
       k=0 k=1 k=2 k=3 k=4 k=5 k=6 k=7
  n=1:  1   1   1   1   1   1   1   1
  n=2:  1   2   3   4   5   6   7   8
  n=3:  1   2   3   4   5   6   7   8
  n=4:  1   3   5   7   9  11  13  15
  n=5:  1   2   3   4   5   6   7   8
  n=6:  1   4   9  16  25  36  49  64
  n=7:  1   2   3   4   5   6   7   8
  n=8:  1   4   7  10  13  16  19  22
  n=9:  1   3   5   7   9  11  13  15
Triangle begins:
  1
  1  1
  1  2  1
  1  3  2  1
  1  4  3  3  1
  1  5  4  5  2  1
  1  6  5  7  3  4  1
  1  7  6  9  4  9  2  1
  1  8  7 11  5 16  3  4  1
  1  9  8 13  6 25  4  7  3  1
  1 10  9 15  7 36  5 10  5  4  1
  1 11 10 17  8 49  6 13  7  9  2  1
  1 12 11 19  9 64  7 16  9 16  3  6  1
  1 13 12 21 10 81  8 19 11 25  4 15  2  1
For example, row n = 8 counts the following divisors:
  1  64  243  256  125  36  7  1
     32  81   128  25   18  1
     16  27   64   5    12
     8   9    32   1    9
     4   3    16        6
     2   1    8         4
     1        4         3
              2         2
              1         1

Seiichi Manyama, Antidiagonals n = 1..140, flattened

Columns k=1..9 of the array give A000005, A048691, A048785, A344327, A344328, A344329, A343526, A344335, A344336.
Row n = 6 of the array is A000290.
Diagonal n = k of the array is A062319.
Array antidiagonal sums (row sums of the triangle) are A343657.
Dominated by A343658.
A000312 = n^n.
A007318 counts k-sets of elements of {1..n}.
A009998(n,k) = n^k (as an array, offset 1).
A059481 counts k-multisets of elements of {1..n}.
Cf. A000169, A000272, A002064, A002109, A066959, A143773, A146291, A176029, A251683, A282935, A326358, A327527, A334996, A343652.

Table[DivisorSigma[0,k^(n-k)],{n,10},{k,n}]

A(n, k) = numdiv(n^k); \\ Seiichi Manyama, May 15 2021

A(n,k) = A000005(A009998(n,k)), where A009998(n,k) = n^k is the interpretation as an array.

A(n,k) = Sum_{d|n} k^omega(d). - Seiichi Manyama, May 15 2021

A051675 "Second factorials": Product_{k=1..n} k^(k^2).

Original entry on oeis.org

1, 16, 314928, 1352605460594688, 403107840000000000000000000000000, 4157825501361955044460594652554199040000000000000000000000000

Offset: 1

N. J. A. Sloane

Spyros G. Kanellos: Mathematical Researches and Results [in Greek]. Athens, 1965.

G. C. Greubel, Table of n, a(n) for n = 1..13
Karim Belabas and Henri Cohen, Numerical Algorithms for Number Theory Using PariGP, American Mathematical Society, 2021, p. 196.

Cf. A002109, A255321, A255323, A255344.

Cf. A243262.

[(&*[k^(k^2): k in [1..n]]): n in [1..10]]; // G. C. Greubel, Oct 14 2018

A051675 := proc(n) local k; mul(k^(k^2),k=1..n); end;

Table[Product[k^(k^2),{k,1,n}],{n,1,10}] (* Vaclav Kotesovec, Feb 20 2015 *)

for(n=1, 10, print1(prod(k=1,n, k^(k^2)), ", ")) \\ G. C. Greubel, Oct 14 2018

a(n) ~ n^(n*(n+1)*(2n+1)/6) / exp(n^3/9 - n/12 - Zeta(3)/(4*Pi^2)), where Zeta(3) = A002117 = 1.2020569031595942853997... . - Vaclav Kotesovec, Feb 20 2015

A255269 a(n) = Product_{k=1..n} k!^k.

Original entry on oeis.org

1, 4, 864, 286654464, 7132880358604800000, 993710590042385551668019200000000000, 82086865668400428790437436119503664712777728000000000000000000

Offset: 1

Vaclav Kotesovec, Feb 20 2015

nonn

Eric Weisstein's World of Mathematics, Hyperfactorial
Eric Weisstein's World of Mathematics, Superfactorial

Cf. A000178, A002109, A036740, A055462, A255268.

Table[Product[k!^k,{k,1,n}],{n,1,10}]
FoldList[Times,Table[(k!)^k,{k,10}]] (* Harvey P. Dale, Aug 16 2021 *)

a(n) = A255268(n) / A055462(n-1).

a(n) ~ sqrt(A) * exp((3 - 45*n^2 - 32*n^3 - 9*Zeta(3)/Pi^2)/72) * n^((8*n^3 + 18*n^2 + 10*n + 1)/24) * (2*Pi)^(n*(n+1)/4), where A = A074962 = 1.28242712910062263687534256886979... is the Glaisher-Kinkelin constant and Zeta(3) = A002117 = 1.2020569031595942853997... .

A255322 a(n) = Product_{k=0..n} (k^2)!.

Original entry on oeis.org

1, 1, 24, 8709120, 182219087869378560000, 2826438545846116156142906806150103040000000000, 1051416277636507481568264360276689674557030810000137484550133942059008000000000000000000

Offset: 0

Vaclav Kotesovec, Feb 21 2015

nonn

Partial products of A088020. - Michel Marcus, Jul 06 2019

Seiichi Manyama, Table of n, a(n) for n = 0..12

Cf. A000178, A255358, A255359, A255360.
Cf. A002109, A051675, A255321, A255323, A255344.
Cf. A000142, A255268, A255269, A255504.
Cf. A272094, A372240, A372241.

Table[Product[(k^2)!, {k, 0, n}], {n, 0, 10}]
FoldList[Times,(Range[0,6]^2)!] (* Harvey P. Dale, Jan 30 2022 *)
Table[(n^2)!^(n+1) / Product[j^(Ceiling[Sqrt[j]]), {j, 1, n^2}], {n, 0, 6}] (* Vaclav Kotesovec, Apr 23 2024 *)
Table[(n^2)!^n * (n!)^2 / Product[j^(Floor[Sqrt[j]]), {j, 1, n^2}], {n, 0, 6}] (* Vaclav Kotesovec, Apr 23 2024 *)

{a(n) = prod(k=1, n, (k^2)!)} \\ Seiichi Manyama, Jul 06 2019

a(n) ~ c * n^((2*n + 1)*(2*n^2 + 2*n + 3)/6) * (2*Pi)^(n/2) / exp(5*n^3/9 + n^2/2 + n), where c = A255504 = 3.048330306522348566911920417337613015885313475... .
From Vaclav Kotesovec, Apr 23 2024: (Start)
a(n) = Product_{j=1..n^2} j^(n - ceiling(sqrt(j)) + 1).
a(n) = (n^2)!^n * (n!)^2 / Product_{j=1..n^2} j^(floor(sqrt(j))). (End)

A255323 Product_{k=1..n} k^(k^4).

Original entry on oeis.org

1, 65536, 29060398333495723291328487792256607374737408

Offset: 1

Vaclav Kotesovec, Feb 21 2015

nonn

The next term a(4) has 198 digits.

Vaclav Kotesovec, Table of n, a(n) for n = 1..5

Cf. A002109, A051675, A255321, A255344, A013663, A243264.

Table[Product[k^(k^4), {k, 1, n}], {n, 1, 5}]

a(n)=prod(k=1,n,k^k^4) \\ Charles R Greathouse IV, Sep 08 2015

a(n) ~ A243264 * n^(n*(n+1)*(2*n+1)*(3*n^2+3*n-1)/30) / exp(n^5/25 - n^3/12 + 13*n/360), where A243264 = exp(-3*Zeta(5)/(4*Pi^4)).

A187059 The exponent of highest power of 2 dividing the product of the elements of the n-th row of Pascal's triangle (A001142).

Original entry on oeis.org

0, 0, 1, 0, 5, 2, 4, 0, 17, 10, 12, 4, 18, 8, 11, 0, 49, 34, 36, 20, 42, 24, 27, 8, 58, 36, 39, 16, 47, 22, 26, 0, 129, 98, 100, 68, 106, 72, 75, 40, 122, 84, 87, 48, 95, 54, 58, 16, 162, 116, 119, 72, 127, 78, 82, 32, 147, 94, 98, 44, 108, 52, 57, 0, 321, 258, 260, 196, 266, 200, 203, 136, 282, 212, 215, 144, 223, 150, 154, 80, 322, 244, 247, 168, 255, 174, 178, 96, 275, 190, 194, 108, 204, 116, 121, 32, 418, 324, 327, 232, 335

Offset: 0

Bruce Reznick, Mar 05 2011

The exponent of the highest power of 2 which divides Product_{k=0..n} binomial(n, k). This can be computed using de Polignac's formula.

This is the function ord_2(Ḡ_n) extensively studied in Lagarias-Mehta (2014), and plotted in Fig. 1.1. - Antti Karttunen, Oct 22 2014

			For example, if n = 4, the power of 2 that divides 1*4*6*4*1 is 5.

I. Niven, H. S. Zuckerman, H. L. Montgomery, An Introduction to the Theory of Numbers, Wiley, 1991, pages 182, 183, 187 (Ex. 34).

Antti Karttunen and Paul Tek, Table of n, a(n) for n = 0..8191 (First 4096 terms from Karttunen)
J. Lagarias, Products of binomial coefficients and Farey fractions, Lecture in DIMACS Conference on Challenges of Identifying Integer Sequences, October 9, 2014; Part 1, Part 2.
Jeffrey C. Lagarias and Harsh Mehta, Products of binomial coefficients and unreduced Farey fractions, arXiv:1409.4145 [math.NT], 2014.

Row sums of triangular table A065040.
Row 1 of array A249421.
Cf. A000295 (a(2^k-2)), A000337 (a(2^k)), A005803 (a(2^k-3)), A036799 (a(2^k+1)), A109363 (a(2^k-4)).
Cf. A000178, A000788, A001142, A002109, A007318, A007814, A174605, A249152, A249150, A249151, A249154, A249343, A249345, A249346, A249347.

a187059 = a007814 . a001142  -- Reinhard Zumkeller, Mar 16 2015

a[n_] := Sum[IntegerExponent[Binomial[n, k], 2], {k, 0, n}]; Array[a, 100, 0]
A187059[n_] := Sum[#*((#+1)*2^k - n - 1) & [Floor[n/2^k]], {k, Floor[Log2[n]]}];
Array[A187059, 100, 0] (* Paolo Xausa, Feb 11 2025 *)
2*Accumulate[#] - Range[Length[#]]*# & [DigitCount[Range[0, 99], 2, 1]] (* Paolo Xausa, Feb 11 2025 *)

a(n)=sum(k=0,n,valuation(binomial(n,k),2))

\\ Much faster version, based on code for A065040 by Charles R Greathouse IV which if reduced even further gives the formula a(n) = 2*A000788(n) - A249154(n):
A065040(m,k) = (hammingweight(k)+hammingweight(m-k)-hammingweight(m));
A187059(n) = sum(k=0, n, A065040(n, k));
for(n=0, 4095, write("b187059.txt", n, " ", A187059(n)));
\\ Antti Karttunen, Oct 25 2014

def A187059(n): return (n+1)*n.bit_count()+sum((m:=1<>j)-(r if n<<1>=m*(r:=k<<1|1) else 0)) for j in range(1,n.bit_length()+1)) # Chai Wah Wu, Nov 11 2024

a(2^k-1) = 0 (19th century); a(2^k) = (k-1)*2^k+1 for k >= 1. (Use de Polignac.)
a(n) = Sum_{i=0..n} A065040(n,i) [where the entries of triangular table A065040(m,k) give the exponent of the maximal power of 2 dividing binomial coefficient A007318(m,k)].
a(n) = A007814(A001142(n)). - Jason Kimberley, Nov 02 2011
a(n) = A249152(n) - A174605(n). [Exponent of 2 in the n-th hyperfactorial minus exponent of 2 in the n-th superfactorial. Cf. for example Lagarias & Mehta paper or Peter Luschny's formula for A001142.] - Antti Karttunen, Oct 25 2014
a(n) = 2*A000788(n) - A249154(n). - Antti Karttunen, Nov 02 2014
a(n) = Sum_{i=1..n} (2*i-n-1)*v_2(i), where v_2(i) = A007814(i) is the exponent of the highest power of 2 dividing i. - Ridouane Oudra, Jun 02 2022
a(n) = Sum_{k=1..floor(log_2(n))} t*((t+1)*2^k - n - 1), where t = floor(n/(2^k)). - Paolo Xausa, Feb 11 2025, derived from Ridouane Oudra's formula above.

Name clarified by Antti Karttunen, Oct 22 2014

A255321 Product_{k=1..n} k^(k^3).

Original entry on oeis.org

1, 256, 1952152956156672, 664283228512699358734362008580999708796213453161234432

Offset: 1

Vaclav Kotesovec, Feb 21 2015

nonn

The next term a(5) has 142 digits.

Vaclav Kotesovec, Table of n, a(n) for n = 1..7

Cf. A002109, A051675, A255323, A255344, A243263.

Table[Product[k^(k^3), {k, 1, n}], {n, 1, 6}]

a(n)=prod(k=2,n,k^k^3) \\ Charles R Greathouse IV, Sep 08 2015

a(n) ~ A243263 * n^(n^2*(n+1)^2/4 - 1/120) / exp(n^4/16 - n^2/12).

A255344 Product_{k=1..n} k^(k^5).

Original entry on oeis.org

1, 4294967296

Offset: 1

Vaclav Kotesovec, Feb 21 2015

nonn

The next terms: a(3) has 126 digits, a(4) has 743 digits, a(5) has 2927 digits.

In general, product_{k=1..n} k^(k^m) ~ A(m) * n^(B(m+1)/(m+1) + sum_{j=1..n} j^m) * exp(-n^(m+1)/(m+1)^2 + sum_{j=1..m-1} (1/(j+1) * B(j+1) * binomial(m,j) * n^(m-j) * sum_{i=0..j-1} 1/(m-i) )), where A(m) is the generalized Glaisher-Kinkelin constant (see A074962, A243262, A243263, A243264, A243265), and B(n) is the Bernoulli number A027641(n) / A027642(n).

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 2.15 Glaisher-Kinkelin constant, p. 137.

Vaclav Kotesovec, Table of n, a(n) for n = 1..4

Cf. A002109, A051675, A255321, A255323, A243265.

Table[Product[k^(k^5), {k, 1, n}], {n, 1, 5}]
FoldList[Times,Table[k^k^5,{k,5}]] (* Harvey P. Dale, Aug 05 2025 *)

a(n)=prod(k=2,n,k^k^5) \\ Charles R Greathouse IV, Sep 08 2015

a(n) ~ A243265 * n^(n^2*(n+1)^2*(2*n^2+2*n-1)/12 + 1/252) / exp(47*n^2/720 - n^4/12 + n^6/36).

A046882 Ultrafactorials: a(n) = n!^n!.

Original entry on oeis.org

1, 1, 4, 46656, 1333735776850284124449081472843776

Offset: 0

Camillo Lamonaca (Camillo.Lamonaca(AT)dva.gov.au)

nonn

a(5) = 3175 042373 780336 892901 667920 556557 182493 442088 021222 004926 225128 381629 943118 937129 098831 435345 716937 405655 305190 657814 877412 786176 000000 000000 000000 000000 000000 000000 000000 000000 000000 000000 000000 000000 000000 000000 000000 000000 000000 000000 000000 000000. - Jonathan Vos Post, Dec 09 2004

Note that, by analogy with factorial primes, subfactorial primes, superfactorial primes and hyperfactorial primes, if a(n)+1 or a(n)-1 is prime, it should be called an ultrafactorial prime. These begin: a(0)+1 = a(1)+1 = 2, a(2)-1 = 3, a(2)+1 = 5. Are there any more? Note that a(3) = 46657 = 13 * 37 * 97 is a 3-brilliant number. a(3)-5, a(3)-3 and a(3)+5 are semiprime; a(3)-7 and a(3)+7 are primes. - Jonathan Vos Post, Dec 09 2004

Jean-Christophe Pain, Bounds on the p-adic valuation of the factorial, hyperfactorial and superfactorial, arXiv:2408.00353 [math.NT], 2024. See p. 6.
Eric Weisstein's World of Mathematics, Ultrafactorial.

Cf. A002109, A100085.

Cf. A000166, A000178, A002982.

lst={};Do[a=n!^n!;AppendTo[lst, a], {n, 6}];lst (* Vladimir Joseph Stephan Orlovsky, Oct 01 2008 *)
#^#&/@(Range[0,5]!) (* Harvey P. Dale, Oct 15 2023 *)

Sum_{n>=1} 1/a(n) = A100085. - Amiram Eldar, Nov 11 2020

A203421 Reciprocal of Vandermonde determinant of (1,1/2,...,1/n).

Original entry on oeis.org

1, 1, -2, -18, 1152, 720000, -5598720000, -658683809280000, 1381360067999170560000, 59463021447701323327733760000, -59463021447701323327733760000000000000, -1542317635347398938581016812202229760000000000000

Offset: 0

Clark Kimberling, Jan 02 2012

Each term divides its successor, as in A000169.

G. C. Greubel, Table of n, a(n) for n = 0..35

Cf. A000169, A002109, A057077, A203422, A203424.

BarnesG:= func< n | (&*[Factorial(k): k in [0..n-2]]) >;
A203421:= func< n | (-1)^Binomial(n,2)*(Factorial(n))^n/BarnesG(n+2) >;
[A203421(n): n in [1..20]]; // G. C. Greubel, Dec 07 2023

(* First program *)
f[j_] := 1/j; z = 12;
v[n_] := Product[Product[f[k] - f[j], {j, 1, k - 1}], {k, 2, n}]
Table[v[n], {n, 1, z}]
1/%  (* A203421 *)
Table[v[n]/v[n + 1], {n, 1, z}]  (* A000169 signed *)
(* Additional programs *)
Table[(-1)^Floor[n/2]*Product[(k + 1)^k, {k, 0, n-1}], {n, 1, 10}] (* Vaclav Kotesovec, Oct 18 2015 *)
Table[(-1)^Binomial[n,2]*(n!)^n/BarnesG[n+2], {n, 20}] (* G. C. Greubel, Dec 07 2023 *)

a(n) = prod(i=2,n, (-i)^(i-1)); \\ Kevin Ryde, Apr 17 2022

def BarnesG(n): return product(factorial(k) for k in range(n-1))
def A203421(n): return (-1)^binomial(n, 2)*(gamma(n+1))^n/BarnesG(n+2)
[A203421(n) for n in range(1, 21)] # G. C. Greubel, Dec 07 2023

G.f.: G(0)/(2*x) -1/x, where G(k)= 1 + 1/(1 - x/(x + (2*k+1)/((2*k+1)^(2*k+1))/(1 + 1/(1 - x/(x - (2*k+2)/((2*k+2)^(2*k+2))/G(k+1)))))); (continued fraction). - Sergei N. Gladkovskii, Jun 03 2013
a(n) = (-1)^floor(n/2) * hyperfactorial(n)/n! = A057077(n) * A002109(n)/n!. - Paul J. Harvey, Feb 08 2014
a(n) = Product_{i=2..n} (-i)^(i-1). - Kevin Ryde, Apr 17 2022
abs(a(n)) ~ A * n^(n*(n-1)/2 - 5/12) / (sqrt(2*Pi) * exp(n^2/4 - n)), where A is the Glaisher-Kinkelin constant A074962. - Vaclav Kotesovec, Nov 20 2023
a(n) = (-1)^binomial(n,2) * (n!)^n / BarnesG(n+2). - G. C. Greubel, Dec 07 2023

a(0)=1 prepended by Alois P. Heinz, Apr 13 2024

cp's OEIS Frontend

A343656 Array read by antidiagonals where A(n,k) is the number of divisors of n^k.

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A051675 "Second factorials": Product_{k=1..n} k^(k^2).

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A255269 a(n) = Product_{k=1..n} k!^k.

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A255322 a(n) = Product_{k=0..n} (k^2)!.

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A255323 Product_{k=1..n} k^(k^4).

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A187059 The exponent of highest power of 2 dividing the product of the elements of the n-th row of Pascal's triangle (A001142).

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A255321 Product_{k=1..n} k^(k^3).

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