A075364 -id:A075364 - cp's OEIS frontend

A075363 Triangle read by rows, in which n-th row gives n smallest powers of n.

1, 2, 4, 3, 9, 27, 4, 16, 64, 256, 5, 25, 125, 625, 3125, 6, 36, 216, 1296, 7776, 46656, 7, 49, 343, 2401, 16807, 117649, 823543, 8, 64, 512, 4096, 32768, 262144, 2097152, 16777216, 9, 81, 729, 6561, 59049, 531441, 4782969, 43046721, 387420489, 10, 100, 1000, 10000, 100000, 1000000, 10000000, 100000000, 1000000000, 10000000000

Offset: 1

Amarnath Murthy, Sep 20 2002

T(n,k) is the number of sequences with repetition (k-tuples) of k (not necessarily different) elements taken from an n-set S. These sequences are also called "words of length k over the alphabet S". For sequences without repetition (partial permutations) cf. A068424. - Manfred Boergens, Jun 18 2023

			From _Felix Fröhlich_, Sep 15 2019: (Start)
Triangle begins:
   1;
   2,   4;
   3,   9,   27;
   4,  16,   64,   256;
   5,  25,  125,   625,   3125;
   6,  36,  216,  1296,   7776,   46656;
   7,  49,  343,  2401,  16807,  117649,  823543;
   8,  64,  512,  4096,  32768,  262144, 2097152, 16777216;
   9,  81,  729,  6561,  59049,  531441, 4782969, 43046721, 387420489; (End)

Paolo Xausa, Table of n, a(n) for n = 1..11325 (rows 1..150 of triangle, flattened).

Cf. A075364, A068424.

T(n, 1) = A000027(n), T(n, n) = A000312(n). Cf. A090414.

Array[#^Range[#] &, 10] (* Paolo Xausa, Jun 09 2025 *)

row(n) = for(k=1, n, print1(n^k, ", "))
trianglerows(n) = for(x=1, n, row(x); print(""))
/* Print initial 10 rows as follows: */
trianglerows(10) \\ Felix Fröhlich, Sep 15 2019

from math import isqrt, comb
def A075363(n): return (isqrt(n<<3)+1>>1)**(n-comb((m:=isqrt(k:=n<<1))+(k>m*(m+1)),2)) # Chai Wah Wu, Jun 08 2025

T(n, k) = n^k, 1<=k<=n.

a(n) = A002024(n)^A002260(n). [Gerald Hillier, Feb 12 2009]

More terms from Antonio G. Astudillo (afg_astudillo(AT)lycos.com), Apr 20 2003

More terms from Michel Marcus, Sep 15 2019

A190343 a(n) = floor(n^((n-1)/2)).

Original entry on oeis.org

1, 1, 3, 8, 25, 88, 343, 1448, 6561, 31622, 161051, 861979, 4826809, 28172943, 170859375, 1073741824, 6975757441, 46753733110, 322687697779, 2289733608959, 16679880978201, 124577080440588, 952809757913927, 7454684703958210, 59604644775390625, 486594112179717592

Offset: 1

Bruno Berselli, May 12 2011

nonn

			a(6) = 88 = floor(6^(5/2)).

Bruno Berselli, Table of n, a(n) for n = 1..200

Cf. A066642, A075364.

Magma
```
[Floor(n^((n-1)/2)): n in [1..24]];
```

Table[Floor[n^((n - 1)/2)], {n, 40}] (* Vincenzo Librandi, Mar 26 2013 *)

for(n=1,20, print1(floor(n^((n-1)/2)), ", ")) \\ G. C. Greubel, Dec 30 2017

from math import isqrt
def A190343(n): return isqrt(n**(n-1)) # Chai Wah Wu, Jun 08 2025

A147771 a(n) = round(n^(n/2)).

Original entry on oeis.org

1, 2, 5, 16, 56, 216, 907, 4096, 19683, 100000, 534146, 2985984, 17403307, 105413504, 661735514, 4294967296, 28761784748, 198359290368, 1406563064942, 10240000000000, 76436817165460, 584318301411328, 4569515072723572

Offset: 1

Vladimir Joseph Stephan Orlovsky, Nov 12 2008

nonn

Chai Wah Wu, Table of n, a(n) for n = 1..500

Cf. A075364, A066642.

lst={};Do[AppendTo[lst,Round[(n^n)^(1/2)]],{n,40}];lst
Table[Round[n^(n/2)],{n,30}] (* Harvey P. Dale, Feb 17 2020 *)

from gmpy2 import isqrt_rem
def A147771(n):
    i, j = isqrt_rem(n**n)
    return int(i+int(4*(j-i) >= 1)) # Chai Wah Wu, Aug 16 2016

cp's OEIS Frontend

A075363 Triangle read by rows, in which n-th row gives n smallest powers of n.

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A190343 a(n) = floor(n^((n-1)/2)).

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A147771 a(n) = round(n^(n/2)).

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