A089792 -id:A089792 - cp's OEIS frontend

A125824 Denominator of n!/3^n.

1, 3, 9, 9, 27, 81, 81, 243, 729, 243, 729, 2187, 2187, 6561, 19683, 19683, 59049, 177147, 59049, 177147, 531441, 531441, 1594323, 4782969, 4782969, 14348907, 43046721, 4782969, 14348907, 43046721, 43046721, 129140163, 387420489

Offset: 0

Benoit Cloitre, Feb 06 2007

G. C. Greubel, Table of n, a(n) for n = 0..1000

A212307 (numerators).

List([0..40], n-> DenominatorRat(Factorial(n)/3^n) ); # G. C. Greubel, Aug 03 2019

[Denominator(Factorial(n)/3^n): n in [0..40]]; // G. C. Greubel, Aug 03 2019

Table[Denominator[n!/3^n], {n,0,40}] (* G. C. Greubel, Aug 03 2019 *)

PARI
```
a(n)=denominator(n!/3^n)
```

[denominator(factorial(n)/3^n) for n in (0..40)] # G. C. Greubel, Aug 03 2019

a(0)=1, a(3n+2) = 3^(n+2)*a(n), a(3n+1) = 3^(n+1)*a(n), a(3n) = 3^n*a(n).
a(n) = 3^A089792(n).
a(n) = denominator((1/(2*Pi)) * Integral_{t=0..2*Pi} exp(i*3*t)(-((Pi-t)/i)^n), i=sqrt(-1). - Paul Barry, Apr 02 2007

A211898 G.f.: exp( Sum_{n>=1} (2^n - (-1)^n)^n * x^n/n ).

Original entry on oeis.org

1, 3, 9, 261, 13419, 7867287, 10444212819, 84955235950827, 2235017786095822257, 273416315791427558035965, 125533366255776787874473759857, 242979442003484538229530424638338553, 1852958949086213206247388599213928431454549

Offset: 0

Paul D. Hanna, Apr 25 2012

nonn

CONJECTURE: the highest power of 3 dividing a(n) equals 3^A089792(n) for n>=0; that is, n!*a(n)/3^n is an integer not divisible by 3 for n>=0.

Given g.f. A(x), note that A(x)^(1/3) is not an integer series.

			G.f.: A(x) = 1 + 3*x + 9*x^2 + 261*x^3 + 13419*x^4 + 7867287*x^5 +...
such that
log(A(x)) = 3*x + 3^2*x^2/2 + 9^3*x^3/3 + 15^4*x^4/4 + 33^5*x^5/5 + 63^6*x^6/6 + 129^7*x^7/7 + 255^8*x^8/8 +...+ (2^n - (-1)^n)^n*x^n/n +...

Cf. A211897, A155200, A089792.

{a(n)=polcoeff(exp(sum(k=1, n, (2^k-(-1)^k)^k*x^k/k)+x*O(x^n)), n)}
for(n=0, 20, print1(a(n), ", "))

a(n) == 3 (mod 6) for n>0.

A353068 Irregular triangle read by rows: T(n,k) = n - multiplicity of prime(k) as a divisor of n!.

Original entry on oeis.org

1, 2, 2, 1, 3, 2, 4, 4, 2, 4, 5, 3, 5, 6, 6, 1, 6, 7, 7, 2, 5, 8, 8, 2, 6, 8, 9, 3, 7, 9, 10, 10, 2, 7, 10, 11, 11, 3, 8, 11, 12, 12, 12, 3, 9, 12, 12, 13, 13, 4, 9, 12, 13, 14, 14, 1, 10, 13, 14, 15, 15, 2, 11, 14, 15, 16, 16, 16, 2, 10, 15, 16, 17, 17, 17, 3, 11, 16, 17, 18, 18, 18, 18

Offset: 2

Michel Marcus, Apr 21 2022

This is the "s" in the American Mathematical Monthly problem.

			First few rows are:
  1;
  2, 2;
  1, 3;
  2, 4, 4;
  2, 4, 5;
  3, 5, 6, 6;
  1, 6, 7, 7;
  2, 5, 8, 8;
  2, 6, 8, 9;
  3, 7, 9, 10, 10;
  2, 7, 10, 11, 11;
  ...

Michel Marcus, Table of n, a(n) for n = 2..10388 (Rows n=2..300).
R. D. Carmichael, Diophantine Analysis: 141, The American Mathematical Monthly, Vol. 14, No. 5 (May, 1907), p. 107.

Cf. A115627, A000120 (column 1), A089792 (column 2).

T[n_, k_] := n - IntegerExponent[n!, Prime[k]]; Table[T[n, k], {n, 2, 19}, {k, 1, PrimePi[n]}] // Flatten (* Amiram Eldar, Apr 21 2022 *)

row(n) = vector(primepi(n), k, n-valuation(n!, prime(k)));

T(n,k) = n - A115627(n, k).

cp's OEIS Frontend

A125824 Denominator of n!/3^n.

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A211898 G.f.: exp( Sum_{n>=1} (2^n - (-1)^n)^n * x^n/n ).

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A353068 Irregular triangle read by rows: T(n,k) = n - multiplicity of prime(k) as a divisor of n!.

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