A368366 -id:A368366 - cp's OEIS frontend

A206344 a(n) = floor(n/2)^n.

0, 1, 1, 16, 32, 729, 2187, 65536, 262144, 9765625, 48828125, 2176782336, 13060694016, 678223072849, 4747561509943, 281474976710656, 2251799813685248, 150094635296999121, 1350851717672992089, 100000000000000000000, 1000000000000000000000, 81402749386839761113321

Offset: 1

Nathaniel Johnston, Feb 06 2012

The sequence gives the number of (potentially unsolvable) "clock puzzles" with n positions in the video game Final Fantasy XIII-2.
Functions from [n] to [n] with f(i) even or f(i) = 1 for all i. - Olivier Gérard, Sep 23 2016
AGM transform of A059841. See A368366 for the definition of the AGM transform. - Alois P. Heinz, Jan 24 2024

G. C. Greubel, Table of n, a(n) for n = 1..425
N. Johnston, Counting and Solving Final Fantasy XIII-2's Clock Puzzles

Cf. A206345, A206346, A276978, A276979 (other classes of endofunctions defined by image parity).

Cf. A059841, A368366.

[Floor(n/2)^n: n in [1..30]]; // G. C. Greubel, Mar 31 2023

Maple
```
seq(floor(n/2)^n, n=1..50);
```
Mathematica
```
Table[Floor[n/2]^n, {n,30}]
```

[(n//2)^n for n in range(1,31)] # G. C. Greubel, Mar 31 2023

A369394 AGM transform of the primes.

Original entry on oeis.org

0, 1, 190, 29761, 9991618, 3349024561, 1787557230622, 1073002497284641, 913569251212186570, 1211439486817121619201, 1701996355944048723430570, 3350440495714062711027347281, 7769260076569386601943106748798, 18992268581018658446853739996365841, 54445901270324824915088660223022735282

Offset: 1

Hugo Pfoertner, Jan 24 2024

See A368366 for the definition of the AGM transform.

Paolo Xausa, Table of n, a(n) for n = 1..195

Cf. A368366.

A369394[n_] := With[{p = Prime[Range[n]]}, Total[p]^n - n^n*Apply[Times, p]];
Array[A369394, 15] (* Paolo Xausa, Jan 29 2024 *)

a369394(n) = {my(v=primes(n)); vecsum(v)^n - n^n*vecprod(v)};

from sympy import prime, primorial
def A369394(n): return sum(prime(i) for i in range(1,n+1))**n-n**n*primorial(n) # Chai Wah Wu, Jan 25 2024

A276978 a(n) = (ceiling(n/2))^n.

Original entry on oeis.org

1, 1, 8, 16, 243, 729, 16384, 65536, 1953125, 9765625, 362797056, 2176782336, 96889010407, 678223072849, 35184372088832, 281474976710656, 16677181699666569, 150094635296999121, 10000000000000000000, 100000000000000000000, 7400249944258160101211

Offset: 1

Olivier Gérard, Sep 23 2016

Functions from [n] to [n] with f(i) odd for all i.
Apart from initial term first differs from A132377 at a(9).
With a(1) = 0: AGM transform of A000035. See A368366 for the definition of the AGM transform. - Alois P. Heinz, Jan 24 2024

Cf. A206344, A276979 (other similar classes of endofunctions).

Cf. A000035, A368366.

Mathematica
```
Table[Ceiling[n/2]^n, {n, 1, 21}]
```

a(n)= ceil(n/2)^n; \\ Michel Marcus, Oct 08 2016

A368367 AGM transform of factorials.

Original entry on oeis.org

0, 1, 405, 1112193, 83733135993, 442674314320893489, 252729042985343953175786217, 20874524971928676951126081870191637441, 321062452616441231930929630956482460885924054669273, 1152310647749051621143585533734466768135769634034830754169423308849

Offset: 1

N. J. A. Sloane, Jan 24 2024

nonn

See A368366 for further information.

Paolo Xausa, Table of n, a(n) for n = 1..30

Cf. A000142, A368366.

A368367[n_] := With[{f = Range[n]!}, Total[f]^n - n^n*Apply[Times, f]];
Array[A368367, 10] (* Paolo Xausa, Jan 29 2024 *)

from math import prod, factorial
def A368367(n): return sum(factorial(i) for i in range(1,n+1))**n-n**n*prod(i**(n-i+1) for i in range(2,n+1)) # Chai Wah Wu, Jan 25 2024

A368371 AGM transform of powers of 2.

Original entry on oeis.org

0, 4, 1016, 547856, 813732832, 3903659417664, 67987041960443776, 4575641535535493153024, 1216334976081196096854162944, 1285452910821757852273429343896576, 5415346123378152397099190627515485911040, 91076602434014222291049466570765323207327092736

Offset: 1

N. J. A. Sloane, Jan 24 2024

nonn

See A368366 for further information.

Paolo Xausa, Table of n, a(n) for n = 1..55

Cf. A368366.

A369394[n_] := (2^(n+1)-2)^n - n^n*2^(n*(n+1)/2);
Array[A369394, 15] (* Paolo Xausa, Jan 29 2024 *)

def A368371(n): return ((1<>1)) # Chai Wah Wu, Jan 25 2024

A368372 a(n) = numerator of AM(n)-HM(n), where AM(n) and HM(n) are the arithmetic and harmonic means of the first n positive integers.

Original entry on oeis.org

0, 1, 4, 29, 111, 103, 472, 2369, 12965, 30791, 197346, 452993, 3337271, 7485915, 4160656, 18358463, 170991927, 124184839, 1278605110, 110351535, 98802055, 211524139, 2595194516, 16562041459, 219589922071, 464651871609, 2207044831642, 4649180818987, 70862100349605, 148699793966557

Offset: 1

N. J. A. Sloane, Jan 24 2024

			0, 1/6, 4/11, 29/50, 111/137, 103/98, 472/363, 2369/1522, 12965/7129, 30791/14762, 197346/83711, 452993/172042, 3337271/1145993, 7485915/2343466, 4160656/1195757, 18358463/4873118, ...

Paolo Xausa, Table of n, a(n) for n = 1..2000

Cf. A102928/A175441, A368366, A368373.

AM:=proc(n) local i; (add(i,i=1..n)/n); end;
HM:=proc(n) local i; (add(1/i,i=1..n)/n)^(-1); end;
s1:=[seq(AM(n)-HM(n),n=1..50)];

A368372[n_] := Numerator[(n+1)/2 - n/HarmonicNumber[n]];
Array[A368372, 35] (* Paolo Xausa, Jan 29 2024 *)

a368372(n) = numerator((n+1)/2 - n/harmonic(n)) \\ Hugo Pfoertner, Jan 25 2024

from fractions import Fraction
from itertools import count, islice
def agen(): # generator of terms
    A = H = 0
    for n in count(1):
        A += n
        H += Fraction(1, n)
        yield ((A*Fraction(1, n) - n/H)).numerator
print(list(islice(agen(), 30))) # Michael S. Branicky, Jan 24 2024

from fractions import Fraction
from sympy import harmonic
def A368372(n): return (Fraction(n+1,2)-Fraction(n,harmonic(n))).numerator # Chai Wah Wu, Jan 25 2024

A368373 a(n) = denominator of AM(n)-HM(n), where AM(n) and HM(n) are the arithmetic and harmonic means of the first n positive integers.

Original entry on oeis.org

1, 6, 11, 50, 137, 98, 363, 1522, 7129, 14762, 83711, 172042, 1145993, 2343466, 1195757, 4873118, 42142223, 28548602, 275295799, 22334054, 18858053, 38186394, 444316699, 2695645910, 34052522467, 68791484534, 312536252003, 630809177806, 9227046511387, 18609365660294, 290774257297357

Offset: 1

N. J. A. Sloane, Jan 24 2024

			0, 1/6, 4/11, 29/50, 111/137, 103/98, 472/363, 2369/1522, 12965/7129, 30791/14762, 197346/83711, 452993/172042, 3337271/1145993, 7485915/2343466, 4160656/1195757, 18358463/4873118, ...

Paolo Xausa, Table of n, a(n) for n = 1..2000

Cf. A102928/A175441, A368366, A368372.

AM:=proc(n) local i; (add(i,i=1..n)/n); end;
HM:=proc(n) local i; (add(1/i,i=1..n)/n)^(-1); end;
s1:=[seq(AM(n)-HM(n),n=1..50)];

A368373[n_] := Denominator[(n+1)/2 - n/HarmonicNumber[n]];
Array[A368373, 35] (* Paolo Xausa, Jan 29 2024 *)

a368373(n) = denominator((n+1)/2 - n/harmonic(n)) \\ Hugo Pfoertner, Jan 25 2024

from fractions import Fraction
from itertools import count, islice
def agen(): # generator of terms
    A = H = 0
    for n in count(1):
        A += n
        H += Fraction(1, n)
        yield ((A*Fraction(1, n) - n/H)).denominator
print(list(islice(agen(), 31))) # Michael S. Branicky, Jan 24 2024

from fractions import Fraction
from sympy import harmonic
def A368373(n): return (Fraction(n+1,2)-Fraction(n,harmonic(n))).denominator # Chai Wah Wu, Jan 25 2024

A368368 AGM transform of squares.

Original entry on oeis.org

0, 9, 1772, 662544, 458284375, 543682781641, 1033215730131200, 2972173255049281536, 12354182867688591966525, 71417932095699615712890625, 556289577698910589026958125056, 5685963330436142993425055664640000, 74579993174727714813743424870936891459

Offset: 1

N. J. A. Sloane, Jan 24 2024

nonn

See A368366 for further information.

Paolo Xausa, Table of n, a(n) for n = 1..160

Cf. A368366.

A368368[n_] := (n*(n+1)*(2*n+1)/6)^n - n^n*n!^2;
Array[A368368, 15] (* Paolo Xausa, Jan 29 2024 *)

from math import factorial
def A368368(n): return (n*(n+1)*((n<<1)+1)//6)**n-n**n*factorial(n)**2 # Chai Wah Wu, Jan 25 2024

A368374 a(n) = smallest k such that AM(k) - GM(k) >= n, where AM(k) and GM(k) are the arithmetic and geometric means of [1,...,k].

Original entry on oeis.org

1, 11, 19, 27, 35, 43, 50, 58, 66, 74, 81, 89, 97, 104, 112, 120, 127, 135, 143, 150, 158, 165, 173, 181, 188, 196, 204, 211, 219, 226, 234, 242, 249, 257, 264, 272, 280, 287, 295, 302, 310, 318, 325, 333, 340, 348, 356, 363, 371, 378, 386, 394, 401, 409, 416

Offset: 0

N. J. A. Sloane, Jan 27 2024, following a suggestion from Don Reble

nonn

The difference d(x) = AM(1,2,3,...,x) - GM(1,2,3,...,x) increases. The first difference of d(x) approaches a limit, 1/2 - 1/e (0.13212...). So we could define a(n) to be the least x such that d(x) >= n. - Don Reble, Jan 27 2024. Which is what I did.

			The values of AM(i)-GM(i) for i = 1, ..., 11 are 0, 0.0857864376269049512, 0.1828794071678603411, 0.2866361605993568152, 0.3948289153026481077, 0.5062048344760910451, 0.6199848408587035501, 0.7356494004968713999, 0.8528337256030871195, 0.9712713118832352378, 1.0907612204156046410, so a(1) = 11.

Chai Wah Wu, Table of n, a(n) for n = 0..10000

Cf. A068996, A368366.

Digits:=20;
AM := proc(n) local i; add(i,i=1..n)/n; end;
GM := proc(n) local i; mul(i,i=1..n)^(1/n); end;
don := proc(n) evalf(AM(n) - GM(n)); end;
a:=[1]; w:=1;
for i from 1 to 300 do
   if don(i) >= w then a:=[op(a),i]; w:=w+1; fi;
od:
a;

from math import factorial
def A368374(n):
    if n == 0: return 1
    m = (n<<1)-1
    kmin, kmax = m, m
    while factorial(kmax)< (kmax-m)**kmax:
        kmax <<= 1
    while True:
        kmid = kmax+kmin>>1
        if factorial(kmid)<Chai Wah Wu, Jan 27 2024

a(39)-a(54) from Alois P. Heinz, Jan 27 2024

A368369 AGM transform of odd numbers.

Original entry on oeis.org

0, 4, 324, 38656, 6812500, 1691793216, 566933589544, 247467140448256, 136744348012840296, 93452709250000000000, 77479910616937022101996, 76677271817228569527975936, 89338843947334074736463717884, 121104748419604219251183463776256, 189040371972603446582336425781250000, 336742459165125951045187297509382291456

Offset: 1

N. J. A. Sloane, Jan 24 2024

nonn

See A368366 for further information.

Paolo Xausa, Table of n, a(n) for n = 1..210

Cf. A005408, A368366.

A368369[n_] := With[{m = n^n}, m*(m-(2*n-1)!!)];
Array[A368369, 20] (* Paolo Xausa, Jan 29 2024 *)

a368369(n) = {my(v=vector(n, i, i+i-1)); vecsum(v)^n - n^n*vecprod(v)}; \\ Hugo Pfoertner, Jan 24 2024

from sympy import factorial2
def A368369(n): return (m:=n**n)*(m-factorial2((n<<1)-1)) # Chai Wah Wu, Jan 25 2024

cp's OEIS Frontend

A206344 a(n) = floor(n/2)^n.

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A369394 AGM transform of the primes.

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A276978 a(n) = (ceiling(n/2))^n.

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A368367 AGM transform of factorials.

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A368371 AGM transform of powers of 2.

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A368372 a(n) = numerator of AM(n)-HM(n), where AM(n) and HM(n) are the arithmetic and harmonic means of the first n positive integers.

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Maple

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A368373 a(n) = denominator of AM(n)-HM(n), where AM(n) and HM(n) are the arithmetic and harmonic means of the first n positive integers.

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Author

Keywords

Examples

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Programs

Maple

Mathematica

PARI

Python

Python

A368368 AGM transform of squares.