A099129 -id:A099129 - cp's OEIS frontend

A097547 a(n) = n^(2^n).

0, 1, 16, 6561, 4294967296, 23283064365386962890625, 63340286662973277706162286946811886609896461828096, 1487815647197611695910312681741273570332356717154798949898498305086387315423300999654757561928633305897036801

Offset: 0

Yasutoshi Kohmoto, Aug 28 2004

nonn

Number of mappings from 2^X to X where X is an n-set.

Vincenzo Librandi, Table of n, a(n) for n = 0..9 (shortened by _N. J. A. Sloane_, Jan 13 2019)
Yasutoshi Kohmoto, The Other Side of Mathematics [broken link?]
Yasutoshi Kohmoto, The Other Side of Mathematics [via Internet Archive Wayback-machine]

Cf. A000290, A020955, A099129, A216992.

[n^(2^n): n in [0..12]]; // Vincenzo Librandi, May 25 2011

a:= n-> (t-> (t@@n)(n))(j-> j^2):
seq(a(n), n=0..7);  # Alois P. Heinz, Sep 29 2023

Table[n^2^n,{n,0,7}] (* Vladimir Joseph Stephan Orlovsky, Mar 01 2010 *)

a(n)=n^(1<Charles R Greathouse IV, May 25 2011

From R. J. Mathar, Apr 23 2007: (Start)
a(6) = A080174(6) - 1.
a(5) = A013737(10) = A013835(6).
a(4) = A000079(32) = A002489(4) = A000215(5)-1.
(End)
Sum_{n>=1} 1/a(n) = A216992. - Amiram Eldar, Nov 19 2020

A117632 Number of 1's required to build n using {+,T} and parentheses, where T(i) = i*(i+1)/2.

Original entry on oeis.org

1, 2, 2, 3, 4, 2, 3, 4, 4, 3, 4, 4, 5, 6, 4, 5, 6, 6, 7, 6, 2, 3, 4, 4, 5, 6, 4, 3, 4, 5, 5, 6, 6, 5, 6, 4, 5, 6, 6, 7, 8, 4, 5, 6, 4, 5, 6, 6, 5, 6, 6, 7, 8, 8, 3, 4, 5, 5, 6, 7, 5, 6, 6, 7, 6, 4, 5, 6, 6, 7, 8, 6, 7, 8, 8, 5, 6, 4, 5, 6, 6, 7, 6, 6, 7, 8, 6, 7, 8, 8, 5, 6, 7, 7, 8, 9, 7, 8, 6, 7, 8, 8

Offset: 1

Jonathan Vos Post, Apr 08 2006

nonn

This problem has the optimal substructure property.

			a(1) = 1 because "1" has a single 1.
a(2) = 2 because "1+1" has two 1's.
a(3) = 2 because 3 = T(1+1) has two 1's.
a(6) = 2 because 6 = T(T(1+1)).
a(10) = 3 because 10 = T(T(1+1)+1).
a(12) = 4 because 12 = T(T(1+1)) + T(T(1+1)).
a(15) = 4 because 15 = T(T(1+1)+1+1).
a(21) = 2 because 21 = T(T(T(1+1))).
a(28) = 3 because 28 = T(T(T(1+1))+1).
a(55) = 3 because 55 = T(T(T(1+1)+1)).

W. A. Beyer, M. L. Stein and S. M. Ulam, The Notion of Complexity. Report LA-4822, Los Alamos Scientific Laboratory of the University of California, Los Alamos, NM, 1971.
R. K. Guy, Unsolved Problems Number Theory, Sect. F26.

Alois P. Heinz, Table of n, a(n) for n = 1..10000
W. A. Beyer, M. L. Stein and S. M. Ulam, The Notion of Complexity. Report LA-4822, Los Alamos Scientific Laboratory of the University of California, Los Alamos, NM, December 1971. [Annotated scanned copy]
R. K. Guy, Some suspiciously simple sequences, Amer. Math. Monthly 93 (1986), 186-190; 94 (1987), 965; 96 (1989), 905.
Ed Pegg, Jr., Integer Complexity.
Eric Weisstein's World of Mathematics, Integer Complexity.
Wikipedia, Optimal substructure.

See also A023361 = number of compositions into sums of triangular numbers, A053614 = numbers that are not the sum of triangular numbers. Iterated triangular numbers: A050536, A050542, A050548, A050909, A007501.

Cf. A000217, A005245, A005520, A003313, A076142, A076091, A061373, A005421, A023361, A053614, A064097, A025280, A003037, A099129, A050536, A050542, A050548, A050909, A007501.

a:= proc(n) option remember; local m; m:= floor (sqrt (n*2));
      if n<3 then n
    elif n=m*(m+1)/2 then a(m)
    else min (seq (a(i)+a(n-i), i=1..floor(n/2)))
      fi
    end:
seq (a(n), n=1..110);  # Alois P. Heinz, Jan 05 2011

a[n_] := a[n] = Module[{m = Floor[Sqrt[n*2]]}, If[n < 3, n, If[n == m*(m + 1)/2, a[m], Min[Table[a[i] + a[n - i], {i, 1, Floor[n/2]}]]]]];
Array[a, 110] (* Jean-François Alcover, Jun 02 2018, from Maple *)

I do not know how many of these entries have been proved to be minimal. - N. J. A. Sloane, Apr 15 2006

Corrected and extended by Alois P. Heinz, Jan 05 2011

A115410 Sequence of iterated sums of squares (1^2+2^2+3^2+...+n^2).

Original entry on oeis.org

1, 55, 349074740, 7458911738724515315524082613205180, 159232823342755035454279356693126603659457648808279391910878167820461916066223383414616137125812767424153893199341493609630

Offset: 1

Hieronymus Fischer, Jan 22 2006

Can be understood as generalized iterated square pyramidal numbers. The growth of the sequence is bounded by O(n^3^n/3^(n/2)). This can be derived from the growth O(n^3/3) of the power two sum (1^2+2^2+3^2+...+n^2) by iteration.

			a(2) = T(T(2)) = T(5) = 55;
a(3) = T(T(T(3))) = T(T(14)) = T(1015) = 349074740.

Cf. A000330, A099129.

t[n_]:=Sum[k^2,{k,n}];Table[Nest[t[#]&,n,n],{n,5}] (* James C. McMahon, Aug 10 2024 *)

Let T(n):=Sum_{k=1..n} k^2; we define a(1):=T(1), a(2):=T(T(2)) etc., a(n):=T(T(T(...(T(n))...))).

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A097547 a(n) = n^(2^n).

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A117632 Number of 1's required to build n using {+,T} and parentheses, where T(i) = i*(i+1)/2.

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A115410 Sequence of iterated sums of squares (1^2+2^2+3^2+...+n^2).

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