A002109 -id:A002109 - cp's OEIS frontend

A046688 Antidiagonals of square array in which k-th row (k>0) is an arithmetic progression of difference 2^(k-1).

1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 4, 5, 5, 1, 1, 5, 7, 9, 9, 1, 1, 6, 9, 13, 17, 17, 1, 1, 7, 11, 17, 25, 33, 33, 1, 1, 8, 13, 21, 33, 49, 65, 65, 1, 1, 9, 15, 25, 41, 65, 97, 129, 129, 1, 1, 10, 17, 29, 49, 81, 129, 193, 257, 257, 1, 1, 11, 19, 33, 57, 97, 161, 257, 385, 513, 513, 1

Offset: 0

N. J. A. Sloane

			From _Gus Wiseman_, May 08 2021: (Start):
Array A(m,n) = 1 + n*2^(m-1) begins:
       n=0: n=1: n=2: n=3: n=4: n=5: n=6: n=7: n=8: n=9:
  m=0:   1    1    1    1    1    1    1    1    1    1
  m=1:   1    2    3    5    9   17   33   65  129  257
  m=2:   1    3    5    9   17   33   65  129  257  513
  m=3:   1    4    7   13   25   49   97  193  385  769
  m=4:   1    5    9   17   33   65  129  257  513 1025
  m=5:   1    6   11   21   41   81  161  321  641 1281
  m=6:   1    7   13   25   49   97  193  385  769 1537
  m=7:   1    8   15   29   57  113  225  449  897 1793
  m=8:   1    9   17   33   65  129  257  513 1025 2049
  m=9:   1   10   19   37   73  145  289  577 1153 2305
Triangle T(n,k) = 1 + (n-k)*2^(k-1) begins:
   1
   1   1
   1   2   1
   1   3   3   1
   1   4   5   5   1
   1   5   7   9   9   1
   1   6   9  13  17  17   1
   1   7  11  17  25  33  33   1
   1   8  13  21  33  49  65  65   1
   1   9  15  25  41  65  97 129 129   1
   1  10  17  29  49  81 129 193 257 257   1
   1  11  19  33  57  97 161 257 385 513 513   1
(End)

G. H. Hardy, A Theorem Concerning the Infinite Cardinal Numbers, Quart. J. Math., 35 (1904), p. 90 = Collected Papers, Vol. VII, p. 430.

Andrew Howroyd, Table of n, a(n) for n = 0..1325

Row sums are A000079.
Diagonal n = m + 1 of the array is A002064.
Diagonal n = m of the array is A005183.
Column m = 1 of the array is A094373.
Diagonal n = m - 1 of the array is A131056.
A002109 gives hyperfactorials (sigma: A260146, omega: A303281).
A009998(k,n) = n^k.
A009999(n,k) = n^k.
A057156 = (2^n)^(2^n).
A062319 counts divisors of n^n.
Cf. A000169, A000272, A000312, A036289, A343656, A343658.

Table[If[k==0,1,n*2^(k-1)+1],{n,0,9},{k,0,9}] (* ARRAY, Gus Wiseman, May 08 2021 *)
Table[If[k==0,1,1+(n-k)*2^(k-1)],{n,0,10},{k,0,n}] (* TRIANGLE, Gus Wiseman, May 08 2021 *)

A(m,n)={if(m>0, 1+n*2^(m-1), 1)}
{ for(m=0, 10, for(n=0, 10, print1(A(m,n), ", ")); print) } \\ Andrew Howroyd, Mar 07 2020

A(m,n) = 1 + n*2^(m-1) for m > 1. - Andrew Howroyd, Mar 07 2020

As a triangle, T(n,k) = A(k,n-k) = 1 + (n-k)*2^(k-1). - Gus Wiseman, May 08 2021

More terms from Larry Reeves (larryr(AT)acm.org), Apr 06 2000

A056584 Solution to (n^2/a(n))^a(n) = gcd(n^n, Product_{k

Original entry on oeis.org

4, 9, 0, 25, 3, 49, 4, 3, 5, 121, 12, 169, 7, 15, 16, 289, 18, 361, 20, 21, 11, 529, 24, 25, 13, 27, 28, 841, 30, 961, 32, 33, 17, 35, 36, 1369, 19, 39, 40, 1681, 42, 1849, 44, 45, 23, 2209, 48, 49, 50, 51, 52, 2809, 54, 55, 56, 57, 29, 3481, 60, 3721, 31, 63, 64, 65

Offset: 2

Henry Bottomley, Jul 03 2000

nonn

			For n = 4, there are no integer solutions of (16/a)^a = 4, though there are two real solutions of about 14.5454938 and 0.3673156945.

Cf. A000312, A002109, A056582, A056583.

a(2) = 4, a(4) = 0, a(8) = 4, a(9) = 3; if p an odd prime: a(p) = p^2 and a(2p) = p; otherwise if n>1: a(n) = n. Apart from n = 4, a(n) = n^2/A056583(n) = log(A056582(n))/log(A056583(n)).

A112999 Partial sums of A036740.

Original entry on oeis.org

1, 5, 221, 331997, 24883531997, 139314094387531997, 82606411393217618227531997, 6984964247224120535022357995827531997, 109110688415578301444592123476429107940843827531997

Offset: 1

Jonathan Vos Post, Jan 03 2006

			a(1) = (1!)^1 = 1^1 = 1.
a(2) = (1!)^1 + (2!)^2 = 1^1 + 2^2 = 1 + 4 = 5.
a(3) = (1!)^1 + (2!)^2 + (3!)^3 = 1^1 + 2^2 + 6^3 = 1 + 4 + 216 = 221.

Seiichi Manyama, Table of n, a(n) for n = 1..30
G. L. Honaker, Jr. and Chris Caldwell, Prime Curios! 331997

Cf. A000142, A000178, A002109, A036740, A339311.

Table[Sum[Product[m^k,{m,1,k}],{k,1,n}],{n,1,10}] (* Vaclav Kotesovec, Nov 01 2014 *)
Accumulate[Table[(n!)^n,{n,10}]] (* Harvey P. Dale, Dec 23 2019 *)

a(n) = sum(k=1, n, k!^k); \\ Michel Marcus, Nov 30 2020

a(n) = Sum_{k=1..n} (k!)^k.
a(n) = Sum_{k=1..n} (A000142(k))^k.
a(n) = Sum_{k=1..n} A036740(k).
a(n) = Sum_{k=1..n} A002109(k) * A000178(k-1).

A260122 a(n) = floor( Product_{k = 1..n} k^(k/2) ).

Original entry on oeis.org

1, 2, 10, 166, 9295, 2007754, 1822022612, 7463004618900, 146894319913813741, 14689431991381374106820, 7846297508164921345697431897, 23428918818620324499511000487089219, 407740674993626332726840969430118771134776

Offset: 1

Ilya Gutkovskiy, Jul 17 2015

nonn

G. C. Greubel, Table of n, a(n) for n = 1..50
Eric Weisstein's World of Mathematics, Hyperfactorial

Cf. A002109, A074962.

Table[Floor[Sqrt[Hyperfactorial[n]]], {n, 1, 12}]

a(n) = sqrtint(prod(k=2,n,k^k)) \\ Charles R Greathouse IV, Jul 17 2015

a(n) = floor(sqrt(A002109(n))) = A000196(A002109(n)).

a(n) ~ sqrt(A)*n^(n*(n+1)/4+1/24)/exp(n^2/8), where A is the Glaisher-Kinkelin constant (A074962). - Ilya Gutkovskiy, Dec 27 2016

A260262 Number of digits of hyperfactorial(hyperfactorial(n)).

Original entry on oeis.org

1, 1, 5, 10703, 1614696745, 28812381422477890, 100652205682053466439353073, 100862590668529143951825397261798321446, 39596172587764149886638486692811308322476202830248047, 7942534398808419809836601901425429825855063583537701822391757140131840

Offset: 0

Matthew Campbell, Jul 21 2015

			Hyperfactorial(Hyperfactorial(1)) = 1.  There is 1 digit in the number 1.  Because of this, a(1) = 1.

Matthew Campbell, Table of n, a(n) for n = 0..27

Cf. A002109, A055642.

Table[Floor[Log[10, Hyperfactorial[Hyperfactorial[n]]]] + 1, {n, 0, 3}]

hyperfactorial(n)=prod(k=2,n,k^k)
first(m)=vector(m,i,#digits(hyperfactorial(hyperfactorial(i)))) \\ Anders Hellström, Aug 29 2015

a(n) = floor(log_10(hyperfactorial(hyperfactorial(n))))+1.

a(n) = A055642(A002109(A002109(n))).

A260299 Numbers k such that hyperfactorial(prime(k)-1) == 1 (mod prime(k)).

Original entry on oeis.org

1, 2, 4, 15, 17, 22, 23, 27, 28, 31, 34, 43, 46, 47, 54, 56, 61, 63, 67, 73, 75, 76, 83, 91, 92, 95, 96, 101, 107, 109, 111, 115, 117, 120, 129, 132, 141, 143, 144, 146, 149, 150, 153, 154, 155, 164, 167, 181, 190, 192, 193, 205, 208, 214, 215, 219, 224, 225

Offset: 1

Matthew Campbell, Jul 22 2015

nonn

			The 4th prime is 7, and the hyperfactorial of 7 is 4031078400000, which is congruent to 1 mod 7. - _Kellen Myers_, Aug 19 2015

Matthew Campbell and Charles R Greathouse IV, Table of n, a(n) for n = 1..10000 (first 2516 terms from Campbell)

Cf. A000040, A002109, A260178, A260297, A260298.

PrimePi[fQ[n_]:= Mod[Hyperfactorial[n - 1], n] == 1; Select[Prime@Range@250, fQ]] (* Vincenzo Librandi, Aug 20 2015 *)

is(n,p=prime(n))=prod(k=2,p-1,Mod(k,p)^k)==1 \\ Charles R Greathouse IV, Aug 29 2015

a(n) = pi(A260298(n)).

A260619 Arithmetic derivative of hyperfactorial(n).

Original entry on oeis.org

0, 0, 4, 216, 165888, 604800000, 48372940800000, 43156963184025600000, 1392410948543163924480000000, 668916177911197542484208831692800000, 8199617664717905359483850194944000000000000000, 2401010998878767104110478543683244630474752000000000000000

Offset: 0

Matthew Campbell, Sep 17 2015

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..37

Cf. A002109, A003415, A068327.

h:= proc(n) option remember; `if`(n=0, 1, h(n-1)* n^n) end:
a:= proc(n) n^n *`if`(n=0, 0,
      a(n-1)+h(n-1)*n*add(i[2]/i[1], i=ifactors(n)[2]))
    end:
seq(a(n), n=0..15);  # Alois P. Heinz, Sep 18 2015

a[n_] := If[n<2, 0, With[{h = Hyperfactorial[n]}, h Sum[{p, e} = pe; e/p, {pe, FactorInteger[h]}]]];
a /@ Range[0, 15] (* Jean-François Alcover, Nov 14 2020 *)

a(n) = A003415(A002109(n)).

a(n) = A002109(n)*A190121(n) (conjectured).

More terms from Alois P. Heinz, Sep 18 2015

A261175 Number of digits of Hyperfactorial(n).

Original entry on oeis.org

1, 1, 1, 3, 5, 8, 13, 19, 26, 35, 45, 56, 69, 84, 100, 117, 137, 158, 180, 204, 231, 258, 288, 319, 352, 387, 424, 463, 503, 546, 590, 636, 684, 734, 786, 840, 897, 955, 1015, 1077, 1141, 1207, 1275, 1345, 1418, 1492, 1568, 1647, 1728, 1811, 1896, 1983, 2072, 2163, 2257, 2352

Offset: 0

Robert G. Wilson v, Aug 20 2015

nonn

Chai Wah Wu, Table of n, a(n) for n = 0..1000

Cf. A002109, A061010, A260262.

Table[1 + Floor@ Log10@ Hyperfactorial@ n, {n, 0, 55}]

a(n)=my(r=1);for(i=1,n+1,r *= i^i);floor(log(r)/log(10))+1 \\ Anders Hellström, Aug 20 2015

vector(60, n, n--; #Str(prod(k=2, n, k^k))) \\ Michel Marcus, Aug 20 2015

A261175_list, n = [], 1
for i in range(100):
    n *= i**i
    A261175_list.append(len(str(n)))  # Chai Wah Wu, Aug 21 2015

a(n) = 1 + floor( log_10( A002109(n))) = A055642(A002109(n)).

A278868 Second series of Hankel determinants based on hyperfactorial/4.

Original entry on oeis.org

1, 1, 6183, 5772211367657472, 76148812142946816440318638031477145600000, 3940613226283843476344831941863494501303228636304800836707599745608602091520000000000

Offset: 0

Karol A. Penson, Nov 29 2016

nonn

It would be useful to know the formula for this sequence.

Cf. A002109, A277829, A278770, A278860.

a:= n-> LinearAlgebra[Determinant](Matrix(n, (i, j)->
        (t-> mul(k^k, k=0..t)/4)(i+j))):
seq(a(n), n=0..6);  # Alois P. Heinz, Nov 29 2016

Table[Det[Table[Hyperfactorial[i + j]/4, {i, n}, {j, n}]], {n, 6}]

A290770 a(n) = Product_{k=1..n} k^(2*k).

Original entry on oeis.org

1, 1, 16, 11664, 764411904, 7464960000000000, 16249593066946560000000000, 11020848942410302096869949440000000000, 3102093199396597590886754340698424229232640000000000, 465607547420733489126893933985879279492195953053596584509440000000000

Offset: 0

Ilya Gutkovskiy, Aug 10 2017

G. C. Greubel, Table of n, a(n) for n = 0..27
Eric Weisstein's World of Mathematics, Hyperfactorial
Index entries for sequences related to factorial numbers

Cf. A000178, A001044, A001818, A002109, A051675, A055209, A061742, A062206, A074962, A184877, A185141, A260122.

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

Table[Product[k^(2 k), {k, 1, n}], {n, 0, 9}]
Table[Hyperfactorial[n]^2, {n, 0, 9}]
Table[n!^(2 n)/BarnesG[n + 1]^2, {n, 0, 9}]

a(n) = prod(k=1, n, k^(2*k)) \\ Felix Fröhlich, Aug 10 2017

a(n) = A002109(n)^2.
a(n) = A185141(n)/A000178(n-1)^2 for n > 0.
a(n) = (n!)^(2*n)/G(n+1)^2, where G() is the Barnes G-function.
a(n) ~ A^2*exp(-n^2/2)*n^(n*(n+1))*n^(1/6), where A is the Glaisher-Kinkelin constant (A074962).

cp's OEIS Frontend

A046688 Antidiagonals of square array in which k-th row (k>0) is an arithmetic progression of difference 2^(k-1).

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A056584 Solution to (n^2/a(n))^a(n) = gcd(n^n, Product_{k

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A112999 Partial sums of A036740.

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A260122 a(n) = floor( Product_{k = 1..n} k^(k/2) ).

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A260262 Number of digits of hyperfactorial(hyperfactorial(n)).

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A260299 Numbers k such that hyperfactorial(prime(k)-1) == 1 (mod prime(k)).

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A260619 Arithmetic derivative of hyperfactorial(n).

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A261175 Number of digits of Hyperfactorial(n).

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