cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-5 of 5 results.

A002489 a(n) = n^(n^2), or (n^n)^n.

Original entry on oeis.org

1, 1, 16, 19683, 4294967296, 298023223876953125, 10314424798490535546171949056, 256923577521058878088611477224235621321607, 6277101735386680763835789423207666416102355444464034512896, 196627050475552913618075908526912116283103450944214766927315415537966391196809
Offset: 0

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Author

N. J. A. Sloane

Keywords

Comments

The number of closed binary operations on a set of order n. Labeled groupoids.
The values of "googol" in base N: "10^100" in base 2 is 2^4=16; "10^100" in base 3 is 3^9=19683, etc. This is N^^3 by the "lower-valued" (left-associative) definition of the hyper4 or tetration operator (see Munafo webpage). - Robert Munafo, Jan 25 2010
n^(n^k) = (((n^n)^n)^...)^n, with k+1 n's, k >= 0. - Daniel Forgues, May 18 2013

Examples

			a(3) = 19683 because (3^3)^3 = 3^(3^2) = 19683.

References

  • John S. Rose, A Course on Group Theory, Camb. Univ. Press, 1978, see p. 6.
  • 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).

Links

Crossrefs

a(n) = A079172(n) + A023814(n) = A079176(n) + A079179(n);
a(n) = A079182(n) + A023813(n) = A079186(n) + A079189(n);
a(n) = A079192(n) + A079195(n) + A079198(n) + A023815(n).
Cf. A002488, A001329, A002488, A023813, A076113, A090588, A000312, A258102.

Programs

Formula

a(n) = [x^(n^2)] 1/(1 - n*x). - Ilya Gutkovskiy, Oct 10 2017
Sum_{n>=1} 1/a(n) = A258102. - Amiram Eldar, Nov 11 2020

A076113 a(n) = n^(n*(n-1)/2).

Original entry on oeis.org

1, 1, 2, 27, 4096, 9765625, 470184984576, 558545864083284007, 19342813113834066795298816, 22528399544939174411840147874772641, 1000000000000000000000000000000000000000000000, 1890591424712781041871514584574319778449301246603238034051
Offset: 0

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Author

Amarnath Murthy, Oct 09 2002

Keywords

Comments

Number of labeled commutative idempotent groupoids with n elements. [edited by Michel Marcus, Jul 10 2025]
Product of terms in n-th row of A076112.

Links

Crossrefs

Cf. A000169, A002489, A006125, A023037, A030257, A076112, A090588.

Programs

  • PARI
    a(n) = n^(n*(n-1)/2); \\ Joerg Arndt, Nov 04 2013

Extensions

More terms from Antonio G. Astudillo (afg_astudillo(AT)lycos.com), Apr 20 2003
a(0)=1 prepended by Alois P. Heinz, Jun 30 2022

A030247 Number of nonisomorphic idempotent groupoids.

Original entry on oeis.org

1, 1, 3, 138, 700688, 794734575200, 307047114275109035760, 61899500454067972015948863454485, 9279375475116928325576506574232168143663715776
Offset: 0

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Author

Christian G. Bower, Feb 15 1998, May 15 1998 and Dec 03 2003

Keywords

Links

Crossrefs

Cf. A001329, A038015, A038018, A090588.

Formula

For a list n(1), n(2), n(3), ..., let fixF[n] = prod{i, j >= 1}(sum{d|[ i, j ]}(d*n(d))^((i, j)*n(i)*n(j)-(i=j)n(i))).
a(n) = sum {1*s_1+2*s_2+...=n} (fix A[s_1, s_2, ...]/(1^s_1*s_1!*2^s_2*s2!*...)) where fixA[s_1, s_2, ...] = prod {i>=j>=1} f(i, j, s_i, s_j) where f(i, j, s_i, s_j) = {i=j} (sum {d|i} (d*s_d))^(i*s_i^2-s_i) or {i != j} (sum {d|lcm(i, j)} (d*s_d))^(2*gcd(i, j)*s_i*s_j)
a(n) asymptotic to (n^(n^2-n))/n! = A090588(n)/A000142(n)

A355561 Number of n-tuples (p_1, p_2, ..., p_n) of positive integers such that p_{i-1} <= p_i <= n^(i-1).

Original entry on oeis.org

1, 1, 2, 24, 3236, 7173370, 330736663032, 382149784071841422, 12983632019302863224103688, 14912674110246473369128526689667934, 654972005961623890774153743504185499487372010, 1228018869478731662593970252736815943512232438560622483276
Offset: 0

Views

Author

Alois P. Heinz, Jul 06 2022

Keywords

Examples

			a(0) = 1: ( ).
a(1) = 1: (1).
a(2) = 2: (1,1), (1,2).
a(3) = 24: (1,1,1), (1,1,2), (1,1,3), (1,1,4), (1,1,5), (1,1,6), (1,1,7), (1,1,8), (1,1,9), (1,2,2), (1,2,3), (1,2,4), (1,2,5), (1,2,6), (1,2,7), (1,2,8), (1,2,9), (1,3,3), (1,3,4), (1,3,5), (1,3,6), (1,3,7), (1,3,8), (1,3,9).

Links

Crossrefs

Main diagonal of A355576.
Cf. A076113, A090588, A107354, A355519.

Programs

  • Maple
    b:= proc(n, k, i) option remember; `if`(n=0, 1,
      add(b(n-1, k, j), j=1..min(i, k^(n-1))))
    end:
a:= n-> b(n$2, infinity):
seq(a(n), n=0..6);
# second Maple program:
b:= proc(n, k) option remember; `if`(n=0, 1, -add(
      b(j, k)*(-1)^(n-j)*binomial(k^j, n-j), j=0..n-1))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..12);

A120929 Partial sums of n^(n^2), A002489.

Original entry on oeis.org

1, 2, 18, 19701, 4294986997, 298023228171940122, 10314424798788558774343889178, 256923577521069192513410265783009965210785, 6277101735386681020759366944276858929512621227473999723681
Offset: 0

Views

Author

Jonathan Vos Post, Aug 18 2006

Keywords

Comments

After 2, can this ever be prime? This is to A001923 Sum k^k, k=1..n, as k^k^k is to k^k.

Examples

			a(0) = 1 because A002489(0) is given formally as 0^0^0 = 1.
a(1) = 2 because 1 + (1^1)^1 = 1 + 1 = 2.
a(2) = 18 because 2 + (2^2)^2 = 2 + 16 = 18.
a(3) = 19701 because 18 + (3^3)^3 = 18 + 19683 = 19701.
a(4) = 4294986997 = 19701 + (4^4)^4 = 19701 + 4294967296.

Links

Crossrefs

Cf. A001923, A002489, A002488, A001329, A002488, A023813, A076113, A090588.

Programs

  • Mathematica
    Accumulate[Join[{1},Table[n^(n^2),{n,9}]]] (* Harvey P. Dale, Apr 10 2014 *)

Formula

a(n) = Sum_{i=0..n} i^(i^2). a(n) = Sum_{i=0..n} (i^i)^i. In this sequence, we formally define 0^0 = 1.

Extensions

More terms from Harvey P. Dale, Apr 10 2014
Showing 1-5 of 5 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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