A113154 -id:A113154 - cp's OEIS frontend

A000178 Superfactorials: product of first n factorials.

1, 1, 2, 12, 288, 34560, 24883200, 125411328000, 5056584744960000, 1834933472251084800000, 6658606584104736522240000000, 265790267296391946810949632000000000, 127313963299399416749559771247411200000000000, 792786697595796795607377086400871488552960000000000000

Offset: 0

N. J. A. Sloane

a(n) is also the Vandermonde determinant of the numbers 1,2,...,(n+1), i.e., the determinant of the (n+1) X (n+1) matrix A with A[i,j] = i^j, 1 <= i <= n+1, 0 <= j <= n. - Ahmed Fares (ahmedfares(AT)my-deja.com), May 06 2001
a(n) = (1/n!) * D(n) where D(n) is the determinant of order n in which the (i,j)-th element is i^j. - Amarnath Murthy, Jan 02 2002
Determinant of S_n where S_n is the n X n matrix S_n(i,j) = Sum_{d|i} d^j. - Benoit Cloitre, May 19 2002
Appears to be det(M_n) where M_n is the n X n matrix with m(i,j) = J_j(i) where J_k(n) denote the Jordan function of row k, column n (cf. A059380(m)). - Benoit Cloitre, May 19 2002
a(2n+1) = 1, 12, 34560, 125411328000, ... is the Hankel transform of A000182 (tangent numbers) = 1, 2, 16, 272, 7936, ...; example: det([1, 2, 16, 272; 2, 16, 272, 7936; 16, 272, 7936, 353792; 272, 7936, 353792, 22368256]) = 125411328000. - Philippe Deléham, Mar 07 2004
Determinant of the (n+1) X (n+1) matrix whose i-th row consists of terms 1 to n+1 of the Lucas sequence U(i,Q), for any Q. When Q=0, the Vandermonde matrix is obtained. - T. D. Noe, Aug 21 2004
Determinant of the (n+1) X (n+1) matrix A whose elements are A(i,j) = B(i+j) for 0 <= i,j <= n, where B(k) is the k-th Bell number, A000110(k) [I. Mezo, JIS 14 (2011) # 11.1.1]. - T. D. Noe, Dec 04 2004
The Hankel transform of the sequence A090365 is A000178(n+1); example: det([1,1,3; 1,3,11; 3,11,47]) = 12. - Philippe Deléham, Mar 02 2005
Theorem 1.3, page 2, of Polynomial points, Journal of Integer Sequences, Vol. 10 (2007), Article 07.3.6, provides an example of an Abelian quotient group of order (n-1) superfactorial, for each positive integer n. The quotient is obtained from sequences of polynomial values. - E. F. Cornelius, Jr. (efcornelius(AT)comcast.net), Apr 09 2007
Starting with offset 1 this is a 'Matryoshka doll' sequence with alpha=1, the multiplicative counterpart to the additive A000292. seq(mul(mul(i,i=alpha..k), k=alpha..n),n=alpha..12). - Peter Luschny, Jul 14 2009
For n>0, a(n) is also the determinant of S_n where S_n is the n X n matrix, indexed from 1, S_n(i,j)=sigma_i(j), where sigma_k(n) is the generalized divisor sigma function: A000203 is sigma_1(n). - Enrique Pérez Herrero, Jun 21 2010
a(n) is the multiplicative Wiener index of the (n+1)-vertex path. Example: a(4)=288 because in the path on 5 vertices there are 3 distances equal to 2, 2 distances equal to 3, and 1 distance equal to 4 (2*2*2*3*3*4=288). See p. 115 of the Gutman et al. reference. - Emeric Deutsch, Sep 21 2011
a(n-1) = Product_{j=1..n-1} j! = V(n) = Product_{1 <= i < j <= n} (j - i) (a Vandermondian V(n), see the Ahmed Fares May 06 2001 comment above), n >= 1, is in fact the determinant of any n X n matrix M(n) with entries M(n;i,j) = p(j-1,x = i), 1 <= i, j <= n, where p(m,x), m >= 0, are monic polynomials of exact degree m with p(0,x) = 1. This is a special x[i] = i choice in a general theorem given in Vein-Dale, p. 59 (written for the transposed matrix M(n;j,x_i) = p(i-1,x_j) = P_i(x_j) in Vein-Dale, and there a_{k,k} = 1, for k=1..n). See the Aug 26 2013 comment under A049310, where p(n,x) = S(n,x) (Chebyshev S). - Wolfdieter Lang, Aug 27 2013
a(n) is the number of monotonic magmas on n elements labeled 1..n with a symmetric multiplication table. I.e., Product(i,j) >= max(i,j); Product(i,j) = Product(j,i). - Chad Brewbaker, Nov 03 2013
The product of the pairwise differences of n+1 integers is a multiple of a(n) [and this does not hold for any k > a(n)]. - Charles R Greathouse IV, Aug 15 2014
a(n) is the determinant of the (n+1) X (n+1) matrix M with M(i,j) = (n+j-1)!/(n+j-i)!, 1 <= i <= n+1, 1 <= j <= n+1. - Stoyan Apostolov, Aug 26 2014
All terms are in A064807 and all terms after a(2) are in A005101. - Ivan N. Ianakiev, Sep 02 2016
Empirical: a(n-1) is the determinant of order n in which the (i,j)-th entry is the (j-1)-th derivative of x^(x+i-1) evaluated at x=1. - John M. Campbell, Dec 13 2016
Empirical: If f(x) is a smooth, real-valued function on an open neighborhood of 0 such that f(0)=1, then a(n) is the determinant of order n+1 in which the (i,j)-th entry is the (j-1)-th derivative of f(x)/((1-x)^(i-1)) evaluated at x=0. - John M. Campbell, Dec 27 2016
Also the automorphism group order of the n-triangular honeycomb rook graph. - Eric W. Weisstein, Jul 14 2017
Is the zigzag Hankel transform of A000182. That is, a(2*n+1) is the Hankel transform of A000182 and a(2*n+2) is the Hankel transform of A000182(n+1). - Michael Somos, Mar 11 2020
Except for n = 0, 1, superfactorial a(n) is never a square (proof in link Mabry and Cormick, FFF 4 p. 349); however, when k belongs to A349079 (see for further information), there exists m, 1 <= m <= k such that a(k) / m! is a square. - Bernard Schott, Nov 29 2021

			a(3) = (1/6)* | 1 1 1 | 2 4 8 | 3 9 27 |
a(7) = n! * a(n-1) = 7! * 24883200 = 125411328000.
a(12) = 1! * 2! * 3! * 4! * 5! * 6! * 7! * 8! * 9! * 10! * 11! * 12!
= 1^12 * 2^11 * 3^10 * 4^9 * 5^8 * 6^7 * 7^6 * 8^5 * 9^4 * 10^3 * 11^2 * 12^1
= 2^56 * 3^26 * 5^11 * 7^6 * 11^2.
G.f. = 1 + x + 2*x^2 + 12*x^3 + 288*x^4 + 34560*x^5 + 24883200*x^6 + ...

Miklos Bona, editor, Handbook of Enumerative Combinatorics, CRC Press, 2015, page 545.
Steven R. Finch, Mathematical Constants, Cambridge, 2003, pp. 135-145.
A. Fletcher, J. C. P. Miller, L. Rosenhead and L. J. Comrie, An Index of Mathematical Tables. Vols. 1 and 2, 2nd ed., Blackwell, Oxford and Addison-Wesley, Reading, MA, 1962, Vol. 1, p. 50.
R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 1990, p. 231.
H. J. Ryser, Combinatorial Mathematics. Mathematical Association of America, Carus Mathematical Monograph 14, 1963, p. 53.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
R. Vein and P. Dale, Determinants and Their Applications in Mathematical Physics, Springer, 1999.

Boris Hostnik, Table of n, a(n) for n = 0..46
Christian Aebi and Grant Cairns, Generalizations of Wilson's Theorem for Double-, Hyper-, Sub-and Superfactorials, The American Mathematical Monthly 122.5 (2015): 433-443.
Andreas B. G. Blobel, On convolution powers of 1/x, arXiv:2203.09519 [math.CO], 2022.
E. F. Cornelius, Jr. and Phill Schultz, Polynomial points , Journal of Integer Sequences, Vol. 10 (2007), Article 07.3.6.
Selden Crary, Factorization of the Determinant of the Gaussian-Covariance Matrix of Evenly Spaced Points Using an Inter-dimensional Multiset Duality, arXiv preprint arXiv:1406.6326 [math.ST], 2014-2019.
N. Destainville, R. Mosseri and F. Bailly, Configurational Entropy of Codimension-One Tilings and Directed Membranes, J. Stat. Phys. 87, Nos 3/4, 697 (1997).
J. East and R. D. Gray, Idempotent generators in finite partition monoids and related semigroups, arXiv preprint arXiv:1404.2359 [math.GR], 2014.
Richard Ehrenborg, The Hankel determinant of exponential polynomials, Amer. Math. Monthly, 107 (2000), 557-560.
William Q. Erickson and Jan Kretschmann, The structure and normalized volume of Monge polytopes, arXiv:2311.07522 [math.CO], 2023. See p. 7.
Steven R. Finch, Glaisher-Kinkelin Constant (gives asymptotic expressions for A002109, A000178) [Broken link]
Steven R. Finch, Glaisher-Kinkelin Constant (gives asymptotic expressions for A002109, A000178) [From the Wayback machine]
Shyam Sunder Gupta, Fascinating Factorials, Exploring the Beauty of Fascinating Numbers, Springer (2025) Ch. 16, 411-442.
Ivan Gutman, Wolfgang Linert, István Lukovits and Željko Tomović, The multiplicative version of the Wiener index, J. Chem. Inf. Comput. Sci., Vol. 40, No. 1 (2000), pp. 113-116.
Brady Haran and Sophie Maclean, What's special about 288?, Numberphile video (2023).
Aoife Hennessy, A Study of Riordan Arrays with Applications to Continued Fractions, Orthogonal Polynomials and Lattice Paths, Ph. D. Thesis, Waterford Institute of Technology, Oct. 2011.
Nick Hobson, Python program for this sequence.
A. M. Ibrahim, Extension of factorial concept to negative numbers, Notes on Number Theory and Discrete Mathematics, Vol. 19, 2013, 2, 30-42.
Pavel L. Krapivsky, Jean-Marc Luck and Kirone Mallick, Quantum return probability of a system of N non-interacting lattice fermions, Journal of Statistical Mechanics: Theory and Experiment, Vol. 2018, No. 2 (2018), 023104; arXiv preprint, arXiv:1710.08178 [cond-mat.mes-hall], 2017-2018.
Jeffrey C. Lagarias and Harsh Mehta, Products of binomial coefficients and unreduced Farey fractions, International Journal of Number Theory, Vol. 12, No. 1 (2016), pp. 57-91; arXiv preprint, arXiv:1409.4145 [math.NT], 2014-2015.
Mogens Esrom Larsen, Wronskian Harmony, Mathematics Magazine, vol. 63, no. 1, 1990, pp. 33-37.
John W. Layman, The Hankel Transform and Some of its Properties, J. Integer Sequences, 4 (2001), #01.1.5.
Rick Mabry and Laura McCormick, Square products of punctured sequences of factorials, Gaz. Aust. Math. Soc., 2009.
Rémy Mosseri and Francis Bailly, Configurational Entropy in Octagonal Tiling Models, Int. J. Mod. Phys. B, Vol. 7, No. 6-7 (1993), pp. 1427-1436.
Rémy Mosseri, F. Bailly and C. Sire, Configurational Entropy in Random Tiling Models, J. Non-Cryst. Solids, Vol. 153-154 (1993), pp. 201-204.
Amarnath Murthy, Miscellaneous Results and Theorems on Smarandache terms and factor partitions, Smarandache Notions Journal, Vol. 11, No. 1-2-3, Spring 2000.
Amarnath Murthy and Charles Ashbacher, Generalized Partitions and Some New Ideas on Number Theory and Smarandache Sequences, Hexis, Phoenix; USA 2005. See Section 3.14.
Christian Radoux, Query 145, Notices Amer. Math. Soc., 25-3 (1978), p. 197.
Christian Radoux, Déterminants de Hankel et théorème de Sylvester, Séminaire Lotharingien de Combinatoire, B28b (1992), 9 pp.
Vignesh Raman, The Generalized Superfactorial, Hyperfactorial and Primorial functions, arXiv:2012.00882 [math.NT], 2020.
Michel Waldschmidt, Schanuel Property for Elliptic and Quasi-Elliptic Functions, arXiv:2504.14041 [math.NT], 2025. Mentions this sequence, see p. 10.
Eric Weisstein's World of Mathematics, Barnes G-Function.
Eric Weisstein's World of Mathematics, Bell Number.
Eric Weisstein's World of Mathematics, Factorial Products.
Eric Weisstein's World of Mathematics, Graph Automorphism.
Eric Weisstein's World of Mathematics, Lucas Sequence.
Eric Weisstein's World of Mathematics, Superfactorial.
Eric Weisstein's World of Mathematics, Vandermonde Determinant.
Index to divisibility sequences.
Index entries for sequences related to factorial numbers.
Index to sequences related to Olympiads and other Mathematical competitions.

Partial products of A000142.
Cf. A002109, A036561, A000292, A098694, A098695, A113271, A087316, A113208, A113231, A113257, A113258, A113320, A113336, A113498, A113173, A113170, A113475, A113492, A113497, A113533, A113534, A113535, A113153, A113154, A113122, A114045, A055462, A137986, A137987.
A002109(n)*A000178(n-1) = (n!)^n = A036740(n) for n >= 1.
A000178 is the Hankel transform (see A001906 for definition) of A000085, A000110, A000296, A005425, A005493, A005494 and A045379. - John W. Layman, Jul 28 2000
Cf. A255322, A255358, A255359, A255360.
Cf. A051675, A255321, A255323, A255344, A287013.
Cf. A348692, A349079.

[&*[Factorial(k): k in [0..n]]: n in [0..20]]; // Bruno Berselli, Mar 11 2015

A000178 := proc(n)
    mul(i!,i=1..n) ;
end proc:
seq(A000178(n),n=0..10) ; # R. J. Mathar, Oct 30 2015

a[0] := 1; a[1] := 1; a[n_] := n!*a[n - 1]; Table[a[n], {n, 1, 12}] (* Stefan Steinerberger, Mar 10 2006 *)
Table[BarnesG[n], {n, 2, 14}] (* Zerinvary Lajos, Jul 16 2009 *)
FoldList[Times,1,Range[20]!] (* Harvey P. Dale, Mar 25 2011 *)
RecurrenceTable[{a[n] == n! a[n - 1], a[0] == 1}, a, {n, 0, 12}] (* Ray Chandler, Jul 30 2015 *)
BarnesG[Range[2, 20]] (* Eric W. Weisstein, Jul 14 2017 *)

A000178(n):=prod(k!,k,0,n)$ makelist(A000178(n),n,0,30); /* Martin Ettl, Oct 23 2012 */

A000178(n)=prod(k=2,n,k!) \\ M. F. Hasler, Sep 02 2007

a(n)=polcoeff(1-sum(k=0, n-1, a(k)*x^k/prod(j=1, k+1, (1+j!*x+x*O(x^n)) )), n) \\ Paul D. Hanna, Oct 02 2013

for(j=1,13, print1(prod(k=1,j,k^(j-k)),", ")) \\ Hugo Pfoertner, Apr 09 2020

A000178_list, n, m = [1], 1,1
for i in range(1,100):
    m *= i
    n *= m
    A000178_list.append(n) # Chai Wah Wu, Aug 21 2015

from math import prod
def A000178(n): return prod(i**(n-i+1) for i in range(2,n+1)) # Chai Wah Wu, Nov 26 2023

def mono_choices(a,b,n)
    n - [a,b].max
end
def comm_mono_choices(n)
    accum =1
    0.upto(n-1) do |i|
        i.upto(n-1) do |j|
            accum = accum * mono_choices(i,j,n)
        end
    end
    accum
end
1.upto(12) do |k|
    puts comm_mono_choices(k)
end # Chad Brewbaker, Nov 03 2013

a(0) = 1, a(n) = n!*a(n-1). - Lee Hae-hwang, May 13 2003, corrected by Ilya Gutkovskiy, Jul 30 2016
a(0) = 1, a(n) = 1^n * 2^(n-1) * 3^(n-2) * ... * n = Product_{r=1..n} r^(n-r+1). - Amarnath Murthy, Dec 12 2003 [Formula corrected by Derek Orr, Jul 27 2014]
a(n) = sqrt(A055209(n)). - Philippe Deléham, Mar 07 2004
a(n) = Product_{i=1..n} Product_{j=0..i-1} (i-j). - Paul Barry, Aug 02 2008
log a(n) = 0.5*n^2*log n - 0.75*n^2 + O(n*log n). - Charles R Greathouse IV, Jan 13 2012
Asymptotic: a(n) ~ exp(zeta'(-1) - 3/4 - (3/4)*n^2 - (3/2)*n)*(2*Pi)^(1/2 + (1/2)*n)*(n+1)^((1/2)*n^2 + n + 5/12). For example, a(100) is approx. 0.270317...*10^6941. (See A213080.) - Peter Luschny, Jun 23 2012
G.f.: 1 + x/(U(0) - x) where U(k) = 1 + x*(k+1)! - x*(k+2)!/U(k+1); (continued fraction). - Sergei N. Gladkovskii, Oct 02 2012
G.f.: G(0)/2, where G(k) = 1 + 1/(1 - 1/(1 + 1/((k+1)!*x*G(k+1)))); (continued fraction). - Sergei N. Gladkovskii, Jun 14 2013
G.f.: 1 = Sum_{n>=0} a(n)*x^n / Product_{k=1..n+1} (1 + k!*x). - Paul D. Hanna, Oct 02 2013
A203227(n+1)/a(n) -> e, as n -> oo. - Daniel Suteu, Jul 30 2016
From Ilya Gutkovskiy, Jul 30 2016: (Start)
a(n) = G(n+2), where G(n) is the Barnes G-function.
a(n) ~ exp(1/12 - n*(3*n+4)/4)*n^(n*(n+2)/2 + 5/12)*(2*Pi)^((n+1)/2)/A, where A is the Glaisher-Kinkelin constant (A074962).
Sum_{n>=0} (-1)^n/a(n) = A137986. (End)
0 = a(n)*a(n+2)^3 + a(n+1)^2*a(n+2)^2 - a(n+1)^3*a(n+3) for all n in Z (if a(-1)=1). - Michael Somos, Mar 11 2020
Sum_{n>=0} 1/a(n) = A287013 = 1/A137987. - Amiram Eldar, Nov 19 2020
a(n) = Wronskian(1, x, x^2, ..., x^n). - Mohammed Yaseen, Aug 01 2023
From Andrea Pinos, Apr 04 2024: (Start)
a(n) = e^(Sum_{k=1..n} (Integral_{x=1..k+1} Psi(x) dx)).
a(n) = e^(Integral_{x=1..n+1} (log(sqrt(2*Pi)) - (x-1/2) + x*Psi(x)) dx).
a(n) = e^(Integral_{x=1..n+1} (log(sqrt(2*Pi)) - (x-1/2) + (n+1)*Psi(x) - log(Gamma(x))) dx).
Psi(x) is the digamma function. (End)

A087316 a(n) = Sum_{k=1..n} prime(k)^prime(n-k+1).

Original entry on oeis.org

4, 17, 84, 545, 7824, 281771, 51540600, 3347558057, 1146374959980, 288113965730819, 529172633067826888, 283453407513524913023, 4122282265785671687518812, 1586581830624893452605127040309, 412109111737176949907195758658736

Offset: 1

Amarnath Murthy, Sep 03 2003

nonn

			Examples from _Jonathan Vos Post_, Jan 06 2006: (Start)
a(1) = 4 because prime(1)^prime(1) = 2^2 = 4.
a(2) = 17 because prime(1)^prime(2) + prime(2)^prime(1) = 2^3 + 3^2 = 17.
a(3) = 84 because 2^5 + 3^3 + 5^2 = 84.
a(4) = 545 = 2^7 + 3^5 + 5^3 + 7^2.
a(5) = 7824 = 2^11 + 3^7 + 5^5 + 7^3 + 11^2.
a(6) = 281771 = 2^13 + 3^11 + 5^7 + 7^5 + 11^3 + 13^2.
a(7) = 51540600 = 2^17 + 3^13 + 5^11 + 7^7 + 11^5 + 13^3 + 17^2.
a(8) = 3347558057 = 2^19 + 3^17 + 5^13 + 7^11 + 11^7 + 13^5 + 17^3 + 19^2.
a(9) = 1146374959980 = 2^23 + 3^19 + 5^17 + 7^13 + 11^11 + 13^7 + 17^5 + 19^3 + 23^2. (End)

T. D. Noe, Table of n, a(n) for n=1..50

Cf. A000040, A005408, A087315, A113122, A113153, A113154.

a:=n->sum(ithprime(k)^ithprime(n-k+1),k=1..n): seq(a(n),n=1..16); # Emeric Deutsch, Apr 13 2005

Table[Sum[Prime[k]^Prime[n - k + 1], {k, 1, n}], {n, 1, 15}] (* Vaclav Kotesovec, Jun 08 2025 *)

a(n) = sum(k=1, n, prime(k)^prime(n-k+1)); \\ Michel Marcus, Aug 20 2019

from sympy import prime
def a(n): return sum(prime(k)**prime(n-k+1) for k in range(1, n+1))
print([a(n) for n in range(1, 16)]) # Michael S. Branicky, Apr 17 2021

More terms from Sam Alexander, Oct 20 2003
Further terms from Emeric Deutsch, Apr 13 2005
Edited by N. J. A. Sloane, Aug 19 2008 at the suggestion of R. J. Mathar

A113231 Ascending descending base exponent transform of triangular numbers (A000217).

Original entry on oeis.org

1, 4, 34, 956, 106721, 75818480, 490656737694, 22960404169011552, 7141530219670856270919, 20319415706020976355219258316, 1104797870481014132439711155738991604

Offset: 1

Jonathan Vos Post, Jan 07 2006

A003101 is the ascending descending base exponent transform of natural numbers A000027. The ascending descending base exponent transform applied to the Fibonacci numbers is A113122; applied to the tribonacci numbers is A113153; applied to the Lucas numbers is A113154. Since the parity of the triangular numbers cycles odd, odd, even, even; the parity of this sequence cycles odd, even, even, even. The smallest prime in this sequence is a(5) = 127601. What is the next prime? What is the first triangular value?

			a(1) = 1 because T(1)^T(1) = 1^1 = 1.
a(2) = 4 because T(1)^T(2) + T(2)^T(1) = 1^3 + 3^1 = 4.
a(3) = 34 = 1^6 + 3^3 + 6^1.
a(4) = 956 = 1^10 + 3^6 + 6^3 + 10^1.
a(5) = 106721 = 1^15 + 3^10 + 6^6 + 10^3 + 15^1.
a(6) = 75818480 = 1^21 + 3^15 + 6^10 + 10^6 + 15^3 + 21^1.
a(7) = 490656737694 = 1^28 + 3^21 + 6^15 + 10^10 + 15^6 + 21^3 + 28^1.
a(8) = 22960404169011552 = 1^36 + 3^28 + 6^21 + 10^15 + 15^10 + 21^6 + 28^3 + 36^1.
a(9) = 7141530219670856270919 = 1^45 + 3^36 + 6^28 + 10^21 + 15^15 + 21^10 + 28^6 + 36^3 + 45^1.

G. C. Greubel, Table of n, a(n) for n = 1..40

Cf. A000217, A005408, A113122, A113153, A113154.

A000217[n_] := Binomial[n + 1, 2]; Table[Sum[A000217[k]^(A000217[n - k + 1]), {k, 1, n}], {n, 1, 10}] (* G. C. Greubel, May 18 2017 *)

for(n=1,10, print1(sum(k=1,n, (binomial(k+1,2))^(binomial(n-k+2,2))), ", ")) \\ G. C. Greubel, May 18 2017

a(n) = Sum_{i=1..n} (T(i))^(T(n-i+1)), where T(n) are the triangle numbers.
a(n) = Sum_{i=1..n} ((i*(i+1)/2))^((n-i+1)*(n-i+2)/2).
a(n) = Sum_{i=1..n} (A000217(i))^(A000217(n-i+1)).
log(a(n)) ~ n^2 * (-1 + 2*LambertW(2^(-3/2)*exp(1/2)*n))^3 / (8*LambertW(2^(-3/2)*exp(1/2)*n)^2). - Vaclav Kotesovec, Jun 07 2025

A113257 Ascending descending base exponent transform of squares (A000290).

Original entry on oeis.org

1, 5, 266, 268722, 4682453347, 2978988815561863, 722638800922610642480852, 22529984108212742763058965679103268, 57286470055793196612331429228839529219232484069

Offset: 1

Jonathan Vos Post, Jan 07 2006

A003101 is the ascending descending base exponent transform of natural numbers A000027. The ascending descending base exponent transform applied to the Fibonacci numbers is A113122; applied to the tribonacci numbers is A113153; applied to the Lucas numbers is A113154. The smallest prime in this sequence is a(2) = 5. What is the next prime? What is the first square value after 1?

			a(1) = 1 because (1^2)^(1^2) = 1^1 = 1.
a(2) = 5 because (1^2)^(4^1) + (4^1)^(1^4) = 1^4 + 4^1 = 5.
a(3) = 266 = 1^9 + 4^4 + 9^1.
a(4) = 268722 = 1^16 + 4^9 + 9^4 + 16^1.
a(5) = 4682453347 = 1^25 + 4^16 + 9^9 + 16^4 + 25^1.
a(6) = 2978988815561863 = 1^36 + 4^25 + 9^16 + 16^9 + 25^4 + 36^1.
a(7) = 722638800922610642480852 = 1^49 + 4^36 + 9^25 + 16^16 + 25^9 + 36^4 + 49^1.
a(8) = 22529984108212742763058965679103268 = 1^64 + 4^49 + 9^36 + 16^25 + 25^16 + 36^9 + 49^4 + 64^1.
a(9) = 57286470055793196612331429228839529219232484069 = 1^81 + 4^64 + 9^49 + 16^36 + 25^25 + 36^16 + 49^9 + 64^4 + 81^1.

G. C. Greubel, Table of n, a(n) for n = 1..30

Cf. A000290, A005408, A113122, A113153, A113154.

Table[Sum[(k^2)^((n - k + 1)^2), {k, 1, n}], {n, 1, 10}] (* G. C. Greubel, May 18 2017 *)

for(n=1,10, print1(sum(k=1,n, (k^2)^((n-k+1)^2) ), ", ")) \\ G. C. Greubel, May 18 2017

a(n) = Sum_{i=1..n} (i^2)^((n-i+1)^2).
a(n) = Sum_{i=1..n} (A000290(i))^(A000290(n-i+1)).
log(a(n)) ~ n^2 * (-1 + 2*LambertW(exp(1/2)*n/2))^3 / (4*LambertW(exp(1/2)*n/2)^2). - Vaclav Kotesovec, Jun 07 2025

a(4) and a(5) corrected by Giovanni Resta, Jun 13 2016

A113271 Ascending descending base exponent transform of 2^n.

Original entry on oeis.org

1, 3, 9, 41, 593, 135457, 8606778433, 36893769626691833985, 680564733921105089459460297630318346497, 231584178474632390853419071752762496470716041121409734167406717963826481922561

Offset: 0

Jonathan Vos Post, Jan 07 2006

A003101 is the ascending descending base exponent transform of natural numbers A000027. The ascending descending base exponent transform applied to the Fibonacci numbers is A113122; applied to the tribonacci numbers is A113153; applied to the Lucas numbers is A113154. The smallest primes in this (always odd) sequence are a(1) = 3, a(3) = 41 and a(5) = 543. What is the next prime?

			a(0) = 1 because (2^0)^(2^0) = 1^1 = 1.
a(1) = 3 = (2^0)^(2^1) + (2^1)^(2^0) = 1^2 + 2^1.
a(2) = 9 = (2^0)^(2^2) + (2^1)^(2^1) + (2^2)^(2^0) = 1^4 + 2^2 + 4^1.
a(3) = 41 = 1^8 + 2^4 + 4^2 + 8^1.
a(4) = 593 = 1^16 + 2^8 + 4^4 + 8^2 + 16^1
a(5) = 135457 = 1^32 + 2^16 + 4^8 + 8^4 + 16^2 + 32^1.
a(6) = 8606778433 = 1^64 + 2^32 + 4^16 + 8^8 + 16^4 + 32^2 + 64^1.
a(7) = 36893769626691833985 = 1^128 + 2^64 + 4^32 + 8^16 + 16^8 + 32^4 + 64^2 + 128^1.
a(8) = 680564733921105089459460297630318346497 = 1^256 + 2^128 + 4^64 + 8^32 + 16^16 + 32^8 + 64^4 + 128^2 + 256^1.

G. C. Greubel, Table of n, a(n) for n = 0..12

Cf. A000079, A005408, A113122, A113153, A113154.

Table[Sum[(2^k)^(2^(n - k)), {k, 0, n}], {n, 0, 10}] (* G. C. Greubel, May 19 2017 *)

for(n=0,5, print1(sum(k=0,n, (2^k)^(2^(n-k))), ", ")) \\ G. C. Greubel, May 19 2017

a(n) = Sum_{i=0..n} (2^i)^(2^(n-i)).
a(n) = Sum_{i=0..n} (2^(n-i))^(2^i).
a(n) = Sum_{i=0..n} (A000079(i))^(A000079(n-i)).
a(n) ~ 2^(2^(n-1) + 1). - Vaclav Kotesovec, Jun 07 2025

a(4) corrected by Giovanni Resta, Jun 13 2016

Formulas corrected by G. C. Greubel, May 19 2017

A113208 Half-fixed-point of ascending descending base exponent transform.

Original entry on oeis.org

1, 1, 2, 4, 10, 44, 1426, 17592187106356

Offset: 1

Jonathan Vos Post, Jan 06 2006

a(9) has 429 digits.

The ascending descending base exponent transform applied to the Fibonacci numbers is A113122; applied to the tribonacci numbers is A113153; applied to the Lucas numbers is A113154. There is no nontrivial integer fixed point of the transform.

			a(2) = 1 because a(1)^a(2) + a(2)^a(1) = 1^1 + 1^1 = 2 and 2/2 = 1.
a(3) = 2 because a(1)^a(3) + a(2)^a(2) + a(3)^a(1) = 1^2 + 1^1 + 2^1 = 4 and 4/2 = 2.
a(4) = 4 because a(1)^a(4) + a(2)^a(3) + a(3)^a(2) + a(4)^a(1) = 1^4 + 1^2 + 2^1 + 4^1 = 8 and 8/2 = 4.
a(5) = 10 because a(1)^a(5) + a(2)^a(4) + a(3)^a(3) + a(4)^a(2) + a(5)^a(1) = 1^10 + 1^4 + 2^2 + 4^1 + 10^1 = 20 and 20/2 = 10.
a(6) = 44 because 1^44 + 1^10 + 2^4 + 4^2 + 10^1 + 44^1 = 88 and 88/2 = 44.
a(7) = (1^1426 + 1^44 + 2^10 + 4^4 + 10^2 + 44^1 + 1426^1)/2 = 1426.
a(8) = (1^17592187106356 + 1^1426 + 2^44 + 4^10 + 10^4 + 44^2 + 1426^1 + 17592187106356^1)/2 = 17592187106356.

Cf. A113122, A113153, A113154.

nmax = 9; a[1] = 1; Do[a[n] = a[n] /. Solve[a[n] == Sum[a[i]^a[n - i + 1], {i, 1, n}]/2, a[n]][[1]], {n, 2, nmax}]; Table[a[n], {n, 1, nmax}] (* Vaclav Kotesovec, Jun 08 2025 *)

a(1) = 1. For n > 1: a(n) = (1/2) * Sum_{i=1..n} a(i)^a(n-i+1).

A113258 Ascending descending base exponent transform of factorials.

Original entry on oeis.org

1, 3, 11, 125, 16824569, 1329227995784915877642188398793079569

Offset: 1

Jonathan Vos Post, Jan 07 2006

A003101 is the ascending descending base exponent transform of natural numbers A000027. The ascending descending base exponent transform applied to the Fibonacci numbers is A113122; applied to the tribonacci numbers is A113153; applied to the Lucas numbers is A113154. The smallest primes in this (always odd) sequence are a(2) = 3 and a(3) = 11. What is the next prime? Is there a nontrivial power after a(4) = 5^3?

			a(1) = 1 because (1!)^(1!) = 1^1 = 1.
a(2) = 3 because (1!)^(2!) + (2!)^(1!) = 1 + 2 = 3.
a(3) = 11 = (1!)^(3!) + (2!)^(2!) + (3!)^(1!) = 1^6 + 2^2 + 6^1 = 11.
a(4) = 125 = (1!)^(4!) + (2!)^(3!) + (3!)^(2!) + (4!)^(1!).
a(6) = 1329227995784915877642188398793079569 = 1^720 + 2^120 + 6^24 + 24^6 + 120^2 + 720^1.
a(7) = 1!^7! + 2!^6! + 3!^5! + 4!^4! + 5!^3! + 6!^2! + 7!^1! has 217 digits.

Cf. A000142, A005408, A113122, A113153, A113154.

Table[Sum[((k)!)^(n - k + 1)!, {k, 1, n}], {n,1,5}] (* G. C. Greubel, May 18 2017 *)

for(n=1,5, print1(sum(k=1,n, (k!)^((n-k+1)!)), ", ")) \\ G. C. Greubel, May 18 2017

a(n) = Sum_{i = 1..n} (i!)^((n-i+1)!).
a(n) = Sum_{i = 1..n} (n-i+1)!^i!.
a(n) = Sum_{i = 1..n} (A000142(i))^(A000142(n-i+1)).
a(n) ~ 2^((n-1)!). - Vaclav Kotesovec, Jun 08 2025

A113320 a(1)=1 and a(n) for n>1 has the smallest positive value such that Sum_{i=1..n} a(i)^a(n-i+1) is prime.

Original entry on oeis.org

1, 1, 1, 2, 2, 4, 4, 4, 6, 2, 6, 4, 18, 6, 4, 20, 6, 30, 4, 40, 30, 8, 18, 16, 40, 128, 24, 40, 58, 194, 78, 84, 56, 56, 72, 112, 98, 300, 444, 54, 978, 1938, 120, 126, 6, 1750

Offset: 1

Jonathan Vos Post, Jan 07 2006

Previous name was: Least integers so ascending descending base exponent transforms all prime.

This is the first sequence submitted as a solution to an "ascending descending base exponent transform inverse problem" where the sequence is iteratively defined such that the transform meets a constraint. The sequence is infinite, but it is hard to characterize the asymptotic cost of adding an n-th term. A003101 is the ascending descending base exponent transform of natural numbers A000027. The ascending descending base exponent transform applied to the Fibonacci numbers is A113122; applied to the tribonacci numbers is A113153; applied to the Lucas numbers is A113154.

			a(1) = 1 by definition.
a(2) = 1 because 1 is the min such that 1^a(2) + a(2)^1 is prime (p=2).
a(3) = 1 because 1 is the min such that 1^a(3) + 1^1 + a(3)^1 is prime (p=5).
a(4) = 2 because 2 is the min such that 1^a(4) + 1^1 + 3^1 + a(4)^1 is prime (p=7).

Cf. A000040, A005408, A113122, A113153, A113154.

inve[w_] := Total[w^Reverse[w]]; a[1] = 1; a[n_] := a[n] = Block[{k = 0}, While[! PrimeQ[ inve@ Append[Array[a, n-1], ++k]]]; k]; Array[a, 46] (* Giovanni Resta, Jun 13 2016 *)

lista(n)={my(a=vector(n)); a[1]=1; print1(1, ", "); for(n=2, #a, my(t=sum(i=2, n-1, a[i]^a[n-i+1])); my(k=1); while(!ispseudoprime(t+1+k), k++); a[n]=k; print1(k, ", "))} \\ Andrew Howroyd, Jan 03 2020

a(1) = 1. For n>1, a(n) = min {k>0: a(1)^k + k^a(1) + Sum_{i=2..n-1} a(i)^a(n-i+1) is prime}.

Corrected and extended by Giovanni Resta, Jun 13 2016

New name from Giovanni Resta, Jan 03 2020

A113336 Least integers, starting with 2, so ascending descending base exponent transforms all prime.

Original entry on oeis.org

2, 1, 6, 6, 18, 12, 18, 42, 288, 108, 180, 1122, 1458, 660

Offset: 1

Jonathan Vos Post, Jan 07 2006

This is the second sequence submitted as a solution to an "ascending descending base exponent transform inverse problem" where the sequence is iteratively defined such that the transform meets a constraint. The sequence is probably infinite, but it is hard to characterize the asymptotic cost of adding an n-th term (the 9th terms is at least 250). A003101 is the ascending descending base exponent transform of natural numbers A000027. The ascending descending base exponent transform applied to the Fibonacci numbers is A113122; applied to the tribonacci numbers is A113153; applied to the Lucas numbers is A113154.

			a(1) = 2 by definition.
a(2) = 1 because 1 is the min such that 2^a(2) + a(2)^2 is prime (p=3).
a(3) = 6 because 6 is the min such that 2^a(3) + 1^1 + a(3)^2 is prime (2^6 + 1^1 + 6^1 = 101).
a(4) = 6 because 2^6 + 1^6 + 6^1 + 6^2 = 107 is prime.
a(5) = 18 because 2^18 + 1^6 + 6^6 + 6^1 + 18^2 = 309131 is prime.
a(6) = 12 because 2^12 + 1^18 + 6^6 + 6^6 + 18^1 + 12^2 = 97571 is prime.
a(7) = 18 because 2^18 + 1^12 + 6^18 + 6^6 + 18^6 + 12^1 + 18^2 = 101559990989777 is prime.
a(8) = 42 because 2^42 + 1^18 + 6^12 + 6^18 + 18^6 + 12^6 + 18^1 + 42^2 = 105960216961847 is prime.
a(9) > 250.

Cf. A000040, A005408, A113122, A113153, A113154.

a(1) = 2. For n>1: a(n) = min {n>0: Sum_{i=1..n} a(i)^a(n-i+1) is prime}.

a(9)-a(14) from Giovanni Resta, Jun 13 2016

A113498 Ascending descending base exponent transform of omega(n) (A001221).

Original entry on oeis.org

1, 2, 3, 4, 6, 7, 8, 9, 13, 12, 14, 15, 21, 19, 24, 21, 29, 28, 30, 28, 40, 35, 41, 42, 46, 41, 53, 44, 59, 52, 61, 55, 79, 55, 69, 66, 86, 70, 90, 73, 94, 93, 91, 81, 121, 88, 114, 103, 123, 95, 137, 102, 138, 122, 132, 114, 168, 121, 144, 145, 159, 137, 180

Offset: 2

Jonathan Vos Post, Jan 10 2006

			Since omega(n) = A001221(n) = 0, 1, 1, 1, 1, 2, 1, 1, 1, 2 and we skip the initial zero term, we have:
a(1) = 1^1 = 1.
a(2) = 1^1 + 1^1 = 2.
a(3) = 1^1 + 1^1 + 1^1 = 3.
a(4) = 1^1 + 1^1 + 1^1 + 1^1 = 4.
a(5) = 1^1 + 1^1 + 1^1 + 1^1 + 2^1 = 6.
a(9) = 1^1 + 1^1 + 1^1 + 1^1 + 2^2 + 1^1 + 1^1 + 1^1 + 2^1 = 13.

G. C. Greubel, Table of n, a(n) for n = 2..5000

Cf. A001221, A113320, A005408, A113122, A113153, A113154, A113336, A113271, A113258, A113257, A113231, A087316, A113208.

Table[Sum[PrimeNu[k]^(PrimeNu[n - k + 2]), {k, 2, n}], {n, 2, 50}] (* G. C. Greubel, May 18 2017 *)

for(n=2,25, print1(sum(k=2,n, omega(k)^(omega(n-k+2))), ", ")) \\ G. C. Greubel, May 18 2017

a(n) = Sum_{i=1..n} (omega(k))^(omega(n-k+2)).

a(n) = Sum_{i=1..n} (A001221(k))^(A001221(n-k+2)).

Corrected and extended by Giovanni Resta, Jun 13 2016

Formulas corrected by G. C. Greubel, May 18 2017

cp's OEIS Frontend

A000178 Superfactorials: product of first n factorials.

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A113257 Ascending descending base exponent transform of squares (A000290).

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A113271 Ascending descending base exponent transform of 2^n.

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A113208 Half-fixed-point of ascending descending base exponent transform.

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