A066642 -id:A066642 - cp's OEIS frontend

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

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

A275945 Numbers n such that the average of different permutations of digits of n is an integer.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 13, 15, 17, 19, 20, 22, 24, 26, 28, 31, 33, 35, 37, 39, 40, 42, 44, 46, 48, 51, 53, 55, 57, 59, 60, 62, 64, 66, 68, 71, 73, 75, 77, 79, 80, 82, 84, 86, 88, 91, 93, 95, 97, 99, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 111

Offset: 1

Altug Alkan, Aug 29 2016

Complement of A273492.
Permutations with a first digit of 0 are included in the average (i.e. 0010 is taken to be 10, 01 is taken to be 1, etc.).
From Robert Israel, Sep 01 2016: (Start)
n such that A002275(A055642(n))*A007953(n) is divisible by A055642(n).
In particular, contains all k-digit numbers if k is in A014950. (End)

			97 is a term because (97+79) is divisible by 2.
100 is a term because (1+10+100) is divisible by 3.
123 is a term because (123+132+213+231+312+321) is divisible by 6.
1001 is not a term because (11+101+110+1001+1010+1100) is not divisible by 6.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A002275, A014950, A045876, A055642, A066642, A007953, A273492.

f:= proc(n) local L, d, s;
  L:= convert(n, base, 10);
  d:= nops(L);
  s:= convert(L, `+`);
  evalb(s*(10^d-1)/9 mod d = 0)
end proc:
select(f, [$1..10000]); # Robert Israel, Sep 01 2016

Select[Range@ 111, IntegerQ@ Mean@ Map[FromDigits, Permutations@ #] &@ IntegerDigits@ # &] (* Michael De Vlieger, Aug 29 2016 *)

A055642(n) = #Str(n);
A007953(n) = sumdigits(n);
for(n=1, 2000, if((((10^A055642(n)-1)/9)*A007953(n)) % A055642(n) == 0, print1(n, ", ")));

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

A273492 Numbers n such that the average of different permutations of digits of n is not an integer.

Original entry on oeis.org

10, 12, 14, 16, 18, 21, 23, 25, 27, 29, 30, 32, 34, 36, 38, 41, 43, 45, 47, 49, 50, 52, 54, 56, 58, 61, 63, 65, 67, 69, 70, 72, 74, 76, 78, 81, 83, 85, 87, 89, 90, 92, 94, 96, 98, 1000, 1001, 1002, 1004, 1005, 1006, 1008, 1009, 1010, 1011, 1013, 1014, 1015, 1017, 1018, 1019, 1020, 1022, 1023, 1024

Offset: 1

Altug Alkan, Aug 29 2016

Complement of A275945.
Permutations with a first digit of 0 are included in the average (i.e. 0010 is taken to be 10, 01 is taken to be 1, etc.).
From Robert Israel, Sep 01 2016: (Start)
n such that A002275(A055642(n))*A007953(n) is not divisible by A055642(n).
In particular, contains no k-digit numbers if k is in A014950. (End)

			12 is a term because (12+21) = 33 is not divisible by 2.
1000 is a term because (1+10+100+1000) = 1111 is not divisible by 4.
123 is not a term because (123+132+213+231+312+321) is divisible by 6.
1001 is a term because (11+101+110+1001+1010+1100) is not divisible by 6.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A002275, A014950, A055642, A066642, A007953, A045876, A055642, A275945.

f:= proc(n) local L,d,s;
  L:= convert(n,base,10);
  d:= nops(L);
  s:= convert(L,`+`);
  evalb(s*(10^d-1)/9 mod d = 0)
end proc:
remove(f, [$1..10000]); # Robert Israel, Sep 01 2016

Select[Range[2^10], ! IntegerQ@ Mean@ Map[FromDigits, Permutations@ #] &@ IntegerDigits@ # &] (* Michael De Vlieger, Aug 29 2016 *)

A055642(n) = #Str(n);
A007953(n) = sumdigits(n);
for(n=1, 2000, if((((10^A055642(n)-1)/9)*A007953(n)) % A055642(n) != 0, print1(n, ", ")));

A255616 Table read by antidiagonals, T(n,k) = floor(sqrt(k^n)), n >= 0, k >=1.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 3, 2, 1, 1, 2, 4, 5, 4, 1, 1, 2, 5, 8, 9, 5, 1, 1, 2, 6, 11, 16, 15, 8, 1, 1, 2, 7, 14, 25, 32, 27, 11, 1, 1, 3, 8, 18, 36, 55, 64, 46, 16, 1, 1, 3, 9, 22, 49, 88, 125, 128, 81, 22, 1, 1, 3, 10, 27, 64, 129, 216, 279, 256, 140, 32, 1, 1, 3, 11, 31, 81, 181, 343, 529, 625, 512, 243, 45, 1

Offset: 0

Kival Ngaokrajang, Feb 28 2015

			See table in the links.

G. C. Greubel, Table of n, a(n) for the first 100 antidaigonals, flattened
Kival Ngaokrajang, Example of table T(n,k), n = 0..12, k = 1..10

Rows(1..10): A000012, A000196, A000027, A000093, A000290, A155013, A000578, A155014, A000583, A238170.
Columns(1..10): A000012, A017910, A017913, A000079, A017919, A017922, A017925, A017928, A000244, A017934.
Diagonals: A190343, A066642, A075264.

T[n_, k_] := Floor[Sqrt[k^n]]; Table[T[k, n + 1 - k], {n, 0, 15}, {k, 0, n}] (* G. C. Greubel, Dec 30 2017 *)

{for(i=1,20,for(n=0,i-1,a=floor(sqrt((i-n)^n));print1(a,", ")))}

T(n,k) = floor(sqrt(k^n)), n >= 0, k >=1.

Terms a(81) onward added by G. C. Greubel, Dec 30 2017

cp's OEIS Frontend

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

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Python

A275945 Numbers n such that the average of different permutations of digits of n is an integer.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

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

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Python

A273492 Numbers n such that the average of different permutations of digits of n is not an integer.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

A255616 Table read by antidiagonals, T(n,k) = floor(sqrt(k^n)), n >= 0, k >=1.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions