A060904 -id:A060904 - cp's OEIS frontend

A254609 Triangle read by rows: T(n,k) = A243757(n)/(A243757(k)*A243757(n-k)).

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 5, 5, 5, 5, 1, 1, 1, 5, 5, 5, 1, 1, 1, 1, 1, 5, 5, 1, 1, 1, 1, 1, 1, 1, 5, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 5, 5, 5, 5, 1, 5, 5, 5, 5, 1, 1, 1, 5, 5, 5, 1, 1, 5, 5, 5, 1, 1, 1, 1, 1, 5, 5, 1, 1, 1, 5, 5

Offset: 0

Tom Edgar, Feb 02 2015

These are the generalized binomial coefficients associated with A060904.
The exponent of T(n,k) is the number of 'carries' that occur when adding k and n-k in base 5 using the traditional addition algorithm.
If T(n,k) != 0 mod 5, then n dominates k in base 5.
A194459(n) = number of ones in row n. - Reinhard Zumkeller, Feb 04 2015

			The first five terms in A060904 are 1, 1, 1, 1, and 5 and so T(4,2) = 1*1*1*1/((1*1)*(1*1))=1 and T(5,3) = 5*1*1*1*1/((1*1*1)*(1*1))=5.
The triangle begins:
1
1, 1
1, 1, 1
1, 1, 1, 1
1, 1, 1, 1, 1
1, 5, 5, 5, 5, 1
1, 1, 5, 5, 5, 1, 1
1, 1, 1, 5, 5, 1, 1, 1
1, 1, 1, 1, 5, 1, 1, 1, 1
1, 1, 1, 1, 1, 1, 1, 1, 1, 1
1, 5, 5, 5, 5, 1, 5, 5, 5, 5, 1
1, 1, 5, 5, 5, 1, 1, 5, 5, 5, 1, 1
1, 1, 1, 5, 5, 1, 1, 1, 5, 5, 1, 1, 1
1, 1, 1, 1, 5, 1, 1, 1, 1, 5, 1, 1, 1, 1
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1

Reinhard Zumkeller, Rows n = 0..124 of triangle, flattened
Tyler Ball, Tom Edgar, and Daniel Juda, Dominance Orders, Generalized Binomial Coefficients, and Kummer's Theorem, Mathematics Magazine, Vol. 87, No. 2, April 2014, pp. 135-143.
Tyler Ball and Daniel Juda, Dominance over N, Rose-Hulman Undergraduate Mathematics Journal, Vol. 13, No. 2, Fall 2013.
Tom Edgar and Michael Z. Spivey, Multiplicative functions, generalized binomial coefficients, and generalized Catalan numbers, Journal of Integer Sequences, Vol. 19 (2016), Article 16.1.6.

Cf. A060904, A243757, A234957, A242849, A082907.

Cf. A194459.

import Data.List (inits)
a254609 n k = a254609_tabl !! n !! k
a254609_row n = a254609_tabl !! n
a254609_tabl = zipWith (map . div)
   a243757_list $ zipWith (zipWith (*)) xss $ map reverse xss
   where xss = tail $ inits a243757_list
-- Reinhard Zumkeller, Feb 04 2015

T(n,k) = A243757(n)/(A243757(k)*A243757(n-k)).
T(n,k) = Product_{i=1..n} A060904(i)/(Product_{i=1..k} A060904(i)*Product_{i=1..n-k} A060904(i)).
T(n,k) = A060904(n)/n*(k/A060904(k)*T(n-1,k-1)+(n-k)/A060904(n-k)*T(n-1,k)).

A339747 a(n) = (5^(valuation(n, 5) + 1) - 1) / 4.

Original entry on oeis.org

1, 1, 1, 1, 6, 1, 1, 1, 1, 6, 1, 1, 1, 1, 6, 1, 1, 1, 1, 6, 1, 1, 1, 1, 31, 1, 1, 1, 1, 6, 1, 1, 1, 1, 6, 1, 1, 1, 1, 6, 1, 1, 1, 1, 6, 1, 1, 1, 1, 31, 1, 1, 1, 1, 6, 1, 1, 1, 1, 6, 1, 1, 1, 1, 6, 1, 1, 1, 1, 6, 1, 1, 1, 1, 31, 1, 1, 1, 1, 6, 1, 1, 1, 1, 6, 1, 1, 1, 1, 6, 1, 1, 1, 1, 6, 1, 1, 1, 1, 31

Offset: 1

Ilya Gutkovskiy, Dec 15 2020

Sum of powers of 5 dividing n.

Denominator of the quotient sigma(5*n) / sigma(n).

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000203, A001620, A038712, A060904, A080278, A088842, A112765, A323921, A339748.

Table[(5^(IntegerExponent[n, 5] + 1) - 1)/4, {n, 1, 100}]
nmax = 100; CoefficientList[Series[Sum[5^k x^(5^k)/(1 - x^(5^k)), {k, 0, Floor[Log[5, nmax]] + 1}], {x, 0, nmax}], x] // Rest

a(n) = (5^(valuation(n, 5) + 1) - 1)/4; \\ Amiram Eldar, Nov 27 2022

G.f.: Sum_{k>=0} 5^k * x^(5^k) / (1 - x^(5^k)).
L.g.f.: -log(Product_{k>=0} (1 - x^(5^k))).
Dirichlet g.f.: zeta(s) / (1 - 5^(1 - s)).
a(n) = sigma(n)/(sigma(5*n) - 5*sigma(n)), where sigma(n) = A000203(n). - Peter Bala, Jun 10 2022
From Amiram Eldar, Nov 27 2022: (Start)
Multiplicative with a(5^e) = (5^(e+1)-1)/4, and a(p^e) = 1 for p != 5.
Sum_{k=1..n} a(k) ~ n*log_5(n) + (1/2 + (gamma - 1)/log(5))*n, where gamma is Euler's constant (A001620). (End)

A264981 Highest power of 9 dividing n.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 9, 1, 1, 1, 1, 1, 1, 1, 1, 9, 1, 1, 1, 1, 1, 1, 1, 1, 9, 1, 1, 1, 1, 1, 1, 1, 1, 9, 1, 1, 1, 1, 1, 1, 1, 1, 9, 1, 1, 1, 1, 1, 1, 1, 1, 9, 1, 1, 1, 1, 1, 1, 1, 1, 9, 1, 1, 1, 1, 1, 1, 1, 1, 9, 1, 1, 1, 1, 1, 1, 1, 1, 81, 1, 1, 1, 1, 1, 1, 1, 1, 9, 1, 1, 1, 1, 1, 1, 1, 1, 9

Offset: 1

Antti Karttunen, Dec 07 2015

The generalized binomial coefficients produced by this sequence provide an analog to Kummer's Theorem using arithmetic in base 9. - Tom Edgar, Feb 02 2016

			Since 18 = 9 * 2, a(18) = 9. Likewise, since 9 does not divide 17, a(17) = 1. - _Tom Edgar_, Feb 02 2016

Antti Karttunen, Table of n, a(n) for n = 1..6561
Tyler Ball, Tom Edgar, and Daniel Juda, Dominance Orders, Generalized Binomial Coefficients, and Kummer's Theorem, Mathematics Magazine, Vol. 87, No. 2, April 2014, pp. 135-143.
Tom Edgar and Michael Z. Spivey, Multiplicative functions, generalized binomial coefficients, and generalized Catalan numbers, Journal of Integer Sequences, Vol. 19 (2016), Article 16.1.6.

Cf. A001620, A264979.

Similar sequences for other bases: A006519 (2), A038500 (3), A234957 (4), A060904 (5), A234959 (6).

Table[9^Length@ TakeWhile[Reverse@ IntegerDigits[n, 9], # == 0 &], {n, 99}] (* Michael De Vlieger, Dec 09 2015 *)
9^Table[IntegerExponent[n, 9], {n, 150}] (* Vincenzo Librandi, Feb 03 2016 *)

a(n) = 9^valuation(n, 9); \\ Michel Marcus, Dec 08 2015

[9^valuation(i, 9) for i in [1..100]] # Tom Edgar, Feb 02 2016

(define (A264981 n) (let loop ((k 9)) (if (not (zero? (modulo n k))) (/ k 9) (loop (* 9 k)))))

a(n) = 9^valuation(n,9). - Tom Edgar, Feb 02 2016
G.f.: x/(1 - x) + 8 * Sum_{k>=1} 9^(k-1)*x^(9^k)/(1 - x^(9^k)). - Ilya Gutkovskiy, Jul 10 2019
From Amiram Eldar, Dec 31 2022: (Start)
Multiplicative with a(3^e) = 3^(2*floor(e/2)), and a(p^e) = 1 if p != 3.
Dirichlet g.f.: zeta(s)*(9^s-1)/(9^s-9).
Sum_{k=1..n} a(k) ~ (4/(9*log(3)))*n*log(n) + (5/9 + 4*(gamma-1)/(9*log(3)))*n, where gamma is Euler's constant (A001620). (End)

Keyword:mult added by Andrew Howroyd, Jul 20 2018

A106740 Triangle read by rows: greatest common divisors of pairs of Fibonacci numbers greater than 1: T(n, k) = gcd(Fibonacci(n), Fibonacci(k)).

Original entry on oeis.org

2, 1, 3, 1, 1, 5, 2, 1, 1, 8, 1, 1, 1, 1, 13, 1, 3, 1, 1, 1, 21, 2, 1, 1, 2, 1, 1, 34, 1, 1, 5, 1, 1, 1, 1, 55, 1, 1, 1, 1, 1, 1, 1, 1, 89, 2, 3, 1, 8, 1, 3, 2, 1, 1, 144, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 233, 1, 1, 1, 1, 13, 1, 1, 1, 1, 1, 1, 377, 2, 1, 5, 2, 1, 1, 2, 5, 1, 2, 1, 1, 610

Offset: 3

Reinhard Zumkeller, May 15 2005

			Triangle begins as:
  2;
  1, 3;
  1, 1, 5;
  2, 1, 1, 8;
  1, 1, 1, 1, 13;
  1, 3, 1, 1,  1, 21;
  2, 1, 1, 2,  1,  1, 34;
  1, 1, 5, 1,  1,  1,  1, 55;
  1, 1, 1, 1,  1,  1,  1,  1, 89;
  2, 3, 1, 8,  1,  3,  2,  1,  1, 144;
  1, 1, 1, 1,  1,  1,  1,  1,  1,   1, 233;
  1, 1, 1, 1, 13,  1,  1,  1,  1,   1,   1, 377;
  2, 1, 5, 2,  1,  1,  2,  5,  1,   2,   1,   1, 610;

G. C. Greubel, Rows n = 3..52 of the triangle, flattened

Cf. A000012, A000045, A010125, A060904, A061347, A093148, A131534.

T[n_, k_]:= GCD[Fibonacci[n], Fibonacci[k]];
Table[T[n, k], {n,3,18}, {k,3,n}]//Flatten (* G. C. Greubel, Sep 11 2021 *)

def T(n,k): return gcd(fibonacci(n), fibonacci(k))
flatten([[T(n,k) for k in (3..n)] for n in (3..18)]) # G. C. Greubel, Sep 11 2021

T(n, k) = gcd(A000045(n), A000045(k)) for n >= 3 and 3 <= k <= n.
T(n, 3) = abs(A061347(n)).
T(n, 4) = A093148(n-1).
T(n, n) = A000045(n).
From G. C. Greubel, Sep 11 2021: (Start)
T(n, 3) = A131534(n-2).
T(n, 5) = A060904(n).
T(n, 6) = A010125(n).
T(n, n-1) = T(n, n-2) = A000012(n).
T(n, n-3) = A093148(n-5).
T(n, n-4) = A093148(n-5).
T(n, n-5) = A060904(n-5).
T(n, n-6) = A010125(n-6). (End)

A244416 6-adic value of 1/n for n >= 1.

Original entry on oeis.org

1, 6, 6, 36, 1, 6, 1, 216, 36, 6, 1, 36, 1, 6, 6, 1296, 1, 36, 1, 36, 6, 6, 1, 216, 1, 6, 216, 36, 1, 6, 1, 7776, 6, 6, 1, 36, 1, 6, 6, 216, 1, 6, 1, 36, 36, 6, 1, 1296, 1, 6, 6, 36, 1, 216, 1, 216, 6, 6, 1, 36, 1, 6, 36, 46656, 1, 6, 1, 36, 6, 6, 1, 216, 1, 6, 6, 36, 1, 6, 1, 1296, 1296, 6, 1, 36, 1, 6

Offset: 1

Wolfdieter Lang, Jun 30 2014

For the definition of 'g-adic value of x', called |x|_g with g an integer >= 2, see the Mahler reference, p. 7. Sometimes also called g-adic absolute value of x. If g is not a prime then this is called a non-archimeden pseudo-valuation. See Mahler, p. 10.

			a(6) = 6^max(1,1) = 6^1 = 6. a(12) = 6^max(2,1) = 6^2 = 36,
a(18) = 6^max(1,2) = 36, a(24) = 6^max(3,1) = 6^3 = 216, ...
a(2) = 6^1 = 6, a(8) = 6^3 = 216, a(14) = 6^1 = 6, ...
a(3) = 6^1 = 6, a(9) = 6^2 = 36, a(15) = 6^1 = 6, ...
a(4) = 6^2 = 36, a(10) = 6^1 = 6, a(16) = 6^4 = 1296, ...

Kurt Mahler, p-adic numbers and their functions, second ed., Cambridge University Press, 1981.

Cf. A244417, A006519 (g=2), A038500 (g=3), A240226 (g=4), A060904 (g=5).

a[n_] := 6^Max[IntegerExponent[n, {2, 3}]]; Array[a, 100] (* Amiram Eldar, Aug 19 2024 *)

a(n) = 6^max(valuation(n, 2), valuation(n, 3)); \\ Amiram Eldar, Aug 19 2024

a(n) = 1 if n == 1 or 5 (mod 6). a(n) = 6^max(A007814(n), A007949(n)) if n == 0 (mod 6), a(n) = 6^A007814(n) if n == 2 or 4 (mod 6), a(n) = 6^A007949(n) if n == 3 (mod 6). The exponents, called f(1/n) in the Mahler reference, are given in A244417(n).

a(n) = 6^A244417(n). - Amiram Eldar, Aug 19 2024

A305720 Square array T(n, k) read by antidiagonals, n > 0 and k > 0; for any prime number p, the p-adic valuation of T(n, k) is the product of the p-adic valuations of n and of k.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 4, 3, 4, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 16, 1, 2, 1, 1, 1, 3, 1, 1, 3, 1, 1, 1, 8, 1, 4, 5, 4, 1, 8, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 9, 64, 1, 6, 1, 64, 9, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 4, 1, 4, 1, 8, 7, 8

Offset: 1

Rémy Sigrist, Jun 09 2018

The array T is completely multiplicative in both parameters.
For any n > 0 and prime number p, T(n, p) is the highest power of p dividing n.
For any function f associating a nonnegative value to any pair of nonnegative values and such that f(0, 0) = 0, we can build an analog of this sequence, say P_f, such that for any prime number p and any n and k > 0 with p-adic valuations i and j, the p-adic valuation of P_f(n, k) equals f(i, j):
   f(i, j)    P_f
   -------    ---
   i * j      T (this sequence)
   i + j      A003991 (product)
   abs(i-j)   A089913
   min(i, j)  A003989 (GCD)
   max(i, j)  A003990 (LCM)
   i AND j    A059895
   i OR j     A059896
   i XOR j    A059897
If log(N) denotes the set {log(n) : n is in N, the set of the positive integers}, one can define a binary operation on log(N): with prime factorizations n = Product p_i^e_i and k = Product p_i^f_i, set log(n) o log(k) = Sum_{i} (e_i*f_i) * log(p_i). o has the premises of a scalar product even if log(N) isn't a vector space. T(n, k) can be viewed as exp(log(n) o log(k)). - Luc Rousseau, Oct 11 2020

			Array T(n, k) begins:
  n\k|    1    2    3    4    5    6    7    8    9   10
  ---+--------------------------------------------------
    1|    1    1    1    1    1    1    1    1    1    1
    2|    1    2    1    4    1    2    1    8    1    2  -> A006519
    3|    1    1    3    1    1    3    1    1    9    1  -> A038500
    4|    1    4    1   16    1    4    1   64    1    4
    5|    1    1    1    1    5    1    1    1    1    5  -> A060904
    6|    1    2    3    4    1    6    1    8    9    2  -> A065331
    7|    1    1    1    1    1    1    7    1    1    1  -> A268354
    8|    1    8    1   64    1    8    1  512    1    8
    9|    1    1    9    1    1    9    1    1   81    1
   10|    1    2    1    4    5    2    1    8    1   10  -> A132741

Cf. A003989, A003990, A003991, A006519, A007947, A038500, A054496, A059895, A059896, A059897, A060904, A065331, A089913, A132741, A268354, A268357.

T[n_, k_] := With[{p = FactorInteger[GCD[n, k]][[All, 1]]}, If[p == {1}, 1, Times @@ (p^(IntegerExponent[n, p] * IntegerExponent[k, p]))]];
Table[T[n-k+1, k], {n, 1, 15}, {k, 1, n}] // Flatten (* Jean-François Alcover, Jun 11 2018 *)

T(n, k) = my (p=factor(gcd(n, k))[,1]); prod(i=1, #p, p[i]^(valuation(n, p[i]) * valuation(k, p[i])))

T(n, k) = T(k, n) (T is commutative).
T(m, T(n, k)) = T(T(m, n), k) (T is associative).
T(n, k) = 1 iff gcd(n, k) = 1.
T(n, n) = A054496(n).
T(n, A007947(n)) = n.
T(n, 1) = 1.
T(n, 2) = A006519(n).
T(n, 3) = A038500(n).
T(n, 4) = A006519(n)^2.
T(n, 5) = A060904(n).
T(n, 6) = A065331(n).
T(n, 7) = A268354(n).
T(n, 8) = A006519(n)^3.
T(n, 9) = A038500(n)^2.
T(n, 10) = A132741(n).
T(n, 11) = A268357(n).

cp's OEIS Frontend

A254609 Triangle read by rows: T(n,k) = A243757(n)/(A243757(k)*A243757(n-k)).

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A339747 a(n) = (5^(valuation(n, 5) + 1) - 1) / 4.

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A264981 Highest power of 9 dividing n.

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A106740 Triangle read by rows: greatest common divisors of pairs of Fibonacci numbers greater than 1: T(n, k) = gcd(Fibonacci(n), Fibonacci(k)).

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A244416 6-adic value of 1/n for n >= 1.

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A305720 Square array T(n, k) read by antidiagonals, n > 0 and k > 0; for any prime number p, the p-adic valuation of T(n, k) is the product of the p-adic valuations of n and of k.

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