A234957 -id:A234957 - cp's OEIS frontend

A323921 a(n) = (4^(valuation(n, 4) + 1) - 1) / 3.

1, 1, 1, 5, 1, 1, 1, 5, 1, 1, 1, 5, 1, 1, 1, 21, 1, 1, 1, 5, 1, 1, 1, 5, 1, 1, 1, 5, 1, 1, 1, 21, 1, 1, 1, 5, 1, 1, 1, 5, 1, 1, 1, 5, 1, 1, 1, 21, 1, 1, 1, 5, 1, 1, 1, 5, 1, 1, 1, 5, 1, 1, 1, 85, 1, 1, 1, 5, 1, 1, 1, 5, 1, 1, 1, 5, 1, 1, 1, 21, 1, 1, 1, 5, 1, 1, 1, 5, 1, 1, 1, 5, 1, 1, 1, 21, 1, 1, 1, 5

Offset: 1

Ilya Gutkovskiy, Dec 15 2020

Sum of powers of 4 dividing n.

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

Cf. A001620, A038712, A080278, A088842, A234957, A235127, A339747, A339748.

Table[(4^(IntegerExponent[n, 4] + 1) - 1)/3, {n, 1, 100}]
nmax = 100; CoefficientList[Series[Sum[4^k x^(4^k)/(1 - x^(4^k)), {k, 0, Floor[Log[4, nmax]] + 1}], {x, 0, nmax}], x] // Rest

a(n) = (4^(valuation(n, 4) + 1) - 1) / 3; \\ Michel Marcus, Jul 09 2022

def A323921(n): return ((1<<((~n&n-1).bit_length()&-2)+2)-1)//3 # Chai Wah Wu, Jul 09 2022

G.f.: Sum_{k>=0} 4^k * x^(4^k) / (1 - x^(4^k)).
L.g.f.: -log(Product_{k>=0} (1 - x^(4^k))).
Dirichlet g.f.: zeta(s) / (1 - 4^(1 - s)).
From Amiram Eldar, Nov 27 2022: (Start)
Multiplicative with a(2^e) = (4^floor((e+2)/2)-1)/3, and a(p^e) = 1 for p != 2.
Sum_{k=1..n} a(k) ~ n*log_4(n) + (1/2 + (gamma - 1)/log(4))*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

A243756 Triangle read by rows: T(n,k) = A242954(n)/(A242954(k) * A242954(n-k)).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 4, 4, 4, 1, 1, 1, 4, 4, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 4, 4, 4, 1, 4, 4, 4, 1, 1, 1, 4, 4, 1, 1, 4, 4, 1, 1, 1, 1, 1, 4, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 4, 4, 4, 1, 4, 4, 4, 1, 4, 4, 4, 1

Offset: 0

Tom Edgar, Jun 09 2014

The exponent of T(n,k) is the number of 'carries' that occur when adding k and n-k in base 4 using the traditional addition algorithm.

If T(n,k) != 0 mod 4, then n dominates k in base 4.

			The triangle begins:
1;
1, 1;
1, 1, 1;
1, 1, 1, 1;
1, 4, 4, 4, 1;
1, 1, 4, 4, 1, 1;
1, 1, 1, 4, 1, 1, 1;
1, 1, 1, 1, 1, 1, 1, 1;
1, 4, 4, 4, 1, 4, 4, 4, 1;
1, 1, 4, 4, 1, 1, 4, 4, 1, 1;
1, 1, 1, 4, 1, 1, 1, 4, 1, 1, 1;
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1;

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. A082907, A234957, A242849, A242954.

m=50
T=[0]+[4^valuation(i, 4) for i in [1..m]]
Table=[[prod(T[1:i+1])/(prod(T[1:j+1])*prod(T[1:i-j+1])) for j in [0..i]] for i in [0..m-1]]
[x for sublist in Table for x in sublist]

T(n,k) = A242954(n)/(A242954(k) * A242954(n-k)).
T(n,k) = Product_{i=1..n} A234957(i)/(Product_{i=1..k} A234957(i)*Product_{i=1..n-k} A234957(i)).
T(n,k) = A234957(n)/n*(k/A234957(k)*T(n-1,k-1)+(n-k)/A234957(n-k)*T(n-1,k)).

A268355 Highest power of 8 dividing n.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 8, 1, 1, 1, 1, 1, 1, 1, 8, 1, 1, 1, 1, 1, 1, 1, 8, 1, 1, 1, 1, 1, 1, 1, 8, 1, 1, 1, 1, 1, 1, 1, 8, 1, 1, 1, 1, 1, 1, 1, 8, 1, 1, 1, 1, 1, 1, 1, 8, 1, 1, 1, 1, 1, 1, 1, 64, 1, 1, 1, 1, 1, 1, 1, 8, 1, 1, 1, 1, 1, 1, 1, 8, 1, 1, 1, 1, 1, 1, 1

Offset: 1

Tom Edgar, Feb 02 2016

The generalized binomial coefficients produced by this sequence provide an analog to Kummer's Theorem using arithmetic in base 8.

			Since 16 = 8 * 2, a(16) = 8. Likewise, since 8 does not divide 15, a(15) = 1.

Antti Karttunen, Table of n, a(n) for n = 1..65537
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, A006519, A038500, A234957, A244413, A234959.

seq(8^floor(padic:-ordp(n,2)/3), n=1..100); # Robert Israel, Feb 03 2016

8^Table[IntegerExponent[n, 8], {n, 150}] (* Vincenzo Librandi, Feb 03 2016 *)

a(n) = 8^valuation(n, 8); \\ Michel Marcus, Feb 05 2016

def A268355(n): return (m:=(~n&n-1))+1>>(m.bit_length()%3) # Chai Wah Wu, Jul 09 2022

Sage
```
[8^valuation(i, 8) for i in [1..100]]
```

a(n) = 8^valuation(n,8).
a(n) = 8^A244413(n).
G.f.: Sum_{m>=0} 8^m * Sum_{j=1..7} x^(j*8^m)/(1-x^(8^(m+1))). - Robert Israel, Feb 03 2016
From Amiram Eldar, Dec 31 2022: (Start)
Multiplicative with a(2^e) = 2^(3*floor(e/3)), and a(p^e) = 1 if p >= 3.
Dirichlet g.f.: zeta(s)*(8^s-1)/(8^s-8).
Sum_{k=1..n} a(k) ~ (7/(24*log(2)))*n*log(n) + (9/16 + 7*(gamma-1)/(24*log(2)))*n, where gamma is Euler's constant (A001620). (End)

Keyword:mult added by Andrew Howroyd, Jul 20 2018

cp's OEIS Frontend

A323921 a(n) = (4^(valuation(n, 4) + 1) - 1) / 3.

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A264981 Highest power of 9 dividing n.

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A243756 Triangle read by rows: T(n,k) = A242954(n)/(A242954(k) * A242954(n-k)).

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A268355 Highest power of 8 dividing n.

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