A002109 -id:A002109 - cp's OEIS frontend

A056582 Highest common factor (or GCD) of n^n and hyperfactorial(n-1), i.e., gcd(n^n, product(k^k) for k < n).

1, 1, 4, 1, 1728, 1, 65536, 19683, 3200000, 1, 8916100448256, 1, 13492928512, 437893890380859375, 18446744073709551616, 1, 39346408075296537575424, 1, 104857600000000000000000000

Offset: 2

Henry Bottomley, Jul 03 2000

Sequence could be defined as: a(2) = 1, a(4) = 4, a(8) = 65536, a(9) = 19683; if p an odd prime: a(p) = 1 and a(2p) = (4p)^p; otherwise if n > 1: a(n) = n^n.

			a(6) = gcd(46656, 86400000) = 1728.

Chai Wah Wu, Table of n, a(n) for n = 2..200
Mohammad K. Azarian, On the Hyperfactorial Function, Hypertriangular Function, and the Discriminants of Certain Polynomials, International Journal of Pure and Applied Mathematics 36(2), 2007, pp. 251-257. MR2312537. Zbl 1133.11012.

from gmpy2 import gcd
A056582_list, n = [], 1
for i in range(2,201):
    m = i**i
    A056582_list.append(int(gcd(n,m)))
    n *= m # Chai Wah Wu, Aug 21 2015

a(n) = GCD(A000312(n), A002109(n-1)).

Except for n = 4, a(n) = A056583(n)^A056584(n) = A056583(n)^(n^2/A056583(n)) = (n^2/A056584(n))^A056584(n).

A056583 Solution to a(n)^(n^2/a(n)) = gcd(n^n, Product_{k

Original entry on oeis.org

1, 1, 0, 1, 12, 1, 16, 27, 20, 1, 12, 1, 28, 15, 16, 1, 18, 1, 20, 21, 44, 1, 24, 25, 52, 27, 28, 1, 30, 1, 32, 33, 68, 35, 36, 1, 76, 39, 40, 1, 42, 1, 44, 45, 92, 1, 48, 49, 50, 51, 52, 1, 54, 55, 56, 57, 116, 1, 60, 1, 124, 63, 64, 65, 66, 1, 68, 69, 70, 1, 72, 1, 148, 75, 76

Offset: 2

Henry Bottomley, Jul 03 2000

nonn

			For n = 4, there are no integer solutions of a^(16/a) = 4, though there are two real solutions of about 1.099997 and 43.55926.

Antti Karttunen, Table of n, a(n) for n = 2..20000

Cf. A000312, A002109, A056582, A056584.

A056583(n) = if(2==n, 1, if(4==n, 0, if(8==n, 16, if(9==n, 27, if(isprime(n), 1, if(!(n%2) && isprime(n/2), 2*n, n)))))); \\ Antti Karttunen, Jan 22 2025

a(2) = 1, a(4) = 0, a(8) = 16, a(9) = 27; if p an odd prime: a(p) = 1 and a(2p) = 4p; otherwise if n>1: a(n) = n. Apart from n = 4, a(n) = n^2/A056584(n) = A056582(n)^(1/A056584(n)).

A219268 Logarithmic derivative of A001142, where A001142(n) = product{k=1..n} k^k/k!.

Original entry on oeis.org

1, 3, 22, 347, 11986, 956334, 184142134, 87903876147, 105736320973732, 323943204887363938, 2547547949361933790328, 51735228018482706470521574, 2726127372514537039881847535054, 374214400937086673452020875815709240, 134262616041282033840675468757467513112522

Offset: 1

Paul D. Hanna, Nov 16 2012

nonn

A001142(n) = hyperfactorial(n)/superfactorial(n) = A002109(n)/A000178(n).

			L.g.f.: L(x) = x + 3*x^2/2 + 22*x^3/3 + 347*x^4/4 + 11986*x^5/5 + 956334*x^6/6 +...
where
exp(L(x)) = 1 + x + 2*x^2 + 9*x^3 + 96*x^4 + 2500*x^5 + 162000*x^6 + 26471025*x^7 + 11014635520*x^8 +...+ A001142(n)*x^n +...

Cf. A001142, A219266, A219268.

nmax=15; Rest[CoefficientList[Series[Log[Sum[Product[j^j/j!,{j,1,k}]*x^k,{k,0,nmax}]],{x,0,nmax}],x] * Range[0,nmax]] (* Vaclav Kotesovec, Jul 10 2015 *)

{a(n)=n*polcoeff(log(sum(k=0,n+1,prod(j=0,k,j^j/j!)*x^k)+x*O(x^n)),n)}
for(n=1,21,print1(a(n),", "))

a(n) ~ A^2 * exp(n^2/2 + n - 1/12) / (n^(n/2 - 2/3) * (2*Pi)^((n+1)/2)), where A = A074962 = 1.2824271291... is the Glaisher-Kinkelin constant. - Vaclav Kotesovec, Jul 10 2015

A260146 Number of positive divisors of hyperfactorial(n).

Original entry on oeis.org

1, 1, 3, 12, 44, 264, 1020, 8160, 19680, 55104, 182784, 2193408, 4608000, 64512000, 210524160, 560849520, 964157040, 17354826720, 32092508448, 641850168960, 1302952210560, 3134374548480, 9806680558080, 235360333393920, 374108929689600, 740882390169600

Offset: 0

Matthew Campbell, Jul 17 2015

			a(2) = sigma(0, hyperfactorial(2)) = sigma(0, 2^2*1^1) = sigma(0, 4). The divisors of 4 are 1, 2, and 4. The number of divisors is a(2) = 3.

Matthew Campbell and Charles R Greathouse IV, Table of n, a(n) for n = 0..1866 (terms 0..677 from Campbell)

Cf. A000005, A002109.

Table[DivisorSigma[0, Hyperfactorial[n]], {n, 0, 200}]

hf(n,p)=my(s); forstep(k=p,n,p, s+=k); if(nhf(n,p)+1, primes([2,n]))) \\ Charles R Greathouse IV, Jul 17 2015

a(n) = A000005(A002109(n)).

A260297 a(n) = prime(n) - (hyperfactorial(prime(n)-1) mod prime(n)).

Original entry on oeis.org

1, 2, 2, 6, 1, 5, 4, 1, 1, 12, 1, 31, 32, 1, 46, 23, 58, 11, 1, 1, 46, 78, 82, 55, 75, 91, 102, 106, 33, 98, 126, 1, 100, 138, 44, 1, 129, 1, 1, 80, 1, 162, 190, 112, 183, 198, 210, 1, 1, 122, 89, 1, 177, 250, 241, 262, 187, 1, 217, 228, 282, 138, 306, 1, 25

Offset: 1

Matthew Campbell, Jul 22 2015

			a(1) = prime(1) - (hyperfactorial(prime(1)-1)) mod prime(1) = 2 - hyperfactorial(2-1) mod (2) = 2 - 1 mod 2 = 2 - 1 = 1.

Matthew Campbell, Table of n, a(n) for n = 1..10000

Cf. A000040, A002109, A260178 (hyperfactorial(prime(n)-1) mod (prime(n))).

Table[Prime[n] - Mod[Hyperfactorial[Prime[n] - 1], Prime[n]], {n, 1, 70}]

a(n,p=prime(n))=lift(-prod(k=1,p-1,Mod(k,p)^k)) \\ Charles R Greathouse IV, Jul 23 2015

a(n) = prime(n) - A260178(n).

A260298 Primes p such that hyperfactorial(p-1) == 1 (mod p).

Original entry on oeis.org

2, 3, 7, 47, 59, 79, 83, 103, 107, 127, 139, 191, 199, 211, 251, 263, 283, 307, 331, 367, 379, 383, 431, 467, 479, 499, 503, 547, 587, 599, 607, 631, 643, 659, 727, 743, 811, 823, 827, 839, 859, 863, 883, 887, 907, 971, 991, 1087, 1151, 1163, 1171, 1259, 1283

Offset: 1

Matthew Campbell, Jul 22 2015

nonn

Does this contain 2 and the entries of A129518? - R. J. Mathar, Aug 07 2015

Primes p such that (p-1)!! == -1 (mod p). - Thomas Ordowski, Jul 26 2016

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

Cf. A000040, A002109, A129518, A260178, A260297, A260299.

fQ[n_] := Mod[Hyperfactorial[n - 1], n] == 1; Select[ Prime@ Range@ 210, fQ] (* Robert G. Wilson v, Aug 06 2015 *)

is(p)=prod(k=2, p-1, Mod(k, p)^k)==1 && isprime(p) \\ Charles R Greathouse IV, Aug 05 2015

a(n) = prime(A260299(n)).

A278860 First series of Hankel determinants based on hyperfactorial.

Original entry on oeis.org

1, 1, 92, 7207016256, 22448940392028699561050505216, 462177945344267713413229252637478222244311831261385379020800000

Offset: 0

Karol A. Penson, Nov 29 2016

nonn

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

Cf. A002109, A277829, A278770.

a:= n-> LinearAlgebra[Determinant](Matrix(n, (i, j)->
        (t-> mul(k^k, k=0..t))(i+j-1))):
seq(a(n), n=0..6);  # Alois P. Heinz, Nov 29 2016

Table[Det[Table[Hyperfactorial[i + j - 1], {i, n}, {j, n}]], {n, 7}]

A299035 a(n) = [x^n] Product_{k=1..n} 1/(1-k^k*x).

Original entry on oeis.org

1, 1, 21, 23980, 4896624249, 327969374429859111, 11123496833223144303532943536, 273486179312859032380857823231575174373792, 6620886635410516590847876477644821623913997428738363459941

Offset: 0

Seiichi Manyama, Feb 01 2018

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..26

Cf. A002109.

Cf. A007820, A298851, A299036.

Table[SeriesCoefficient[Product[1/(1 - k^k*x), {k, 1, n}], {x, 0, n}], {n, 0, 10}] (* Vaclav Kotesovec, Feb 02 2018 *)

a(n) ~ n^(n^2). - Vaclav Kotesovec, Feb 02 2018

A307084 Expansion of 1/(1 - 1^1x/(1 - 2^2x/(1 - 3^3x/(1 - 4^4x/(1 - 5^5*x/(1 - ...)))))), a continued fraction.

Original entry on oeis.org

1, 1, 5, 133, 31769, 95375641, 4353388262525, 3536446917781244413, 58773633134246903294470769, 22612364832863674279489837434733681, 224919094724957152626614652086970769074005045, 63900685361274641827300282511815586348785532532913331893

Offset: 0

Ilya Gutkovskiy, Mar 23 2019

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..37

Cf. A000312, A002109, A285380.

nmax = 11; CoefficientList[Series[1/(1 + ContinuedFractionK[-k^k x, 1, {k, 1, nmax}]), {x, 0, nmax}], x]

a(n) = my(A=1+O(x)); for(i=1, n, A=1-(n-i+1)^(n-i+1)*x/A); polcoef(1/A, n); \\ Seiichi Manyama, Apr 15 2021

a(n) ~ A002109(n). - Vaclav Kotesovec, Apr 15 2021

A330716 n-th Gosper hyperfactorial of n.

Original entry on oeis.org

1, 1, 16, 1952152956156672

Offset: 0

Greg Huber, Dec 27 2019

Gosper's m-th hyperfactorial of n is the product 1^(1^m)*2^(2^m)*3^(3^m)*...*n^(n^m).

The 0th hyperfactorial is the factorial function.

			n=3: a(3) = 1^(1^3)*2^(2^3)*3^(3^3) = 2^8 * 3^27.
a(4) has 198 decimal digits and a(5) has 2927 digits.

R. W. Gosper, "Fac Fun" (ca. 1979).

R. W. Gosper, Gosper's facfun document

Cf. A000142, A002109, A051675, A255321, A255323, A255344 (0th through 5th Gosper hyperfactorials, respectively).

nmax:=3; Table[Product[i^(i^n),{i,1,n}],{n,0,nmax}] (* Stefano Spezia, Dec 29 2019 *)

cp's OEIS Frontend

A056582 Highest common factor (or GCD) of n^n and hyperfactorial(n-1), i.e., gcd(n^n, product(k^k) for k < n).

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A056583 Solution to a(n)^(n^2/a(n)) = gcd(n^n, Product_{k

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A219268 Logarithmic derivative of A001142, where A001142(n) = product{k=1..n} k^k/k!.

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A260146 Number of positive divisors of hyperfactorial(n).

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A260297 a(n) = prime(n) - (hyperfactorial(prime(n)-1) mod prime(n)).

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A260298 Primes p such that hyperfactorial(p-1) == 1 (mod p).

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A278860 First series of Hankel determinants based on hyperfactorial.

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Mathematica

A299035 a(n) = [x^n] Product_{k=1..n} 1/(1-k^k*x).

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A307084 Expansion of 1/(1 - 1^1x/(1 - 2^2x/(1 - 3^3x/(1 - 4^4x/(1 - 5^5*x/(1 - ...)))))), a continued fraction.