A038500 -id:A038500 - cp's OEIS frontend

A263272 Self-inverse permutation of nonnegative integers: a(n) = A263273(2*n) / 2.

0, 1, 2, 3, 4, 5, 6, 11, 8, 9, 10, 7, 12, 13, 14, 15, 32, 23, 18, 29, 20, 33, 38, 17, 24, 35, 26, 27, 28, 19, 30, 37, 16, 21, 34, 25, 36, 31, 22, 39, 40, 41, 42, 95, 68, 45, 86, 59, 96, 113, 50, 69, 104, 77, 54, 83, 56, 87, 110, 47, 60, 101, 74, 99, 92, 65, 114, 119, 44, 51, 98, 71, 72, 89, 62, 105, 116, 53, 78, 107, 80, 81

Offset: 0

Antti Karttunen, Dec 05 2015

Antti Karttunen, Table of n, a(n) for n = 0..9841
Index entries for sequences that are permutations of the natural numbers

Cf. A000035, A263273.
Bisections: A264986, A264987.
Cf. also A246200, A264967, A264968, A264974, A264978, A264975, A264976, A264984.

f[n_] := Block[{g, h}, g[x_] := x/3^IntegerExponent[x, 3]; h[x_] := x/g@ x; If[n == 0, 0, FromDigits[Reverse@ IntegerDigits[#, 3], 3] &@ g[n] h[n]]]; Table[f[2 n]/2, {n, 0, 81}] (* Michael De Vlieger, Jan 04 2016,after Jean-François Alcover at A263273 *)

from sympy import factorint
from sympy.ntheory.factor_ import digits
from operator import mul
def a030102(n): return 0 if n==0 else int(''.join(map(str, digits(n, 3)[1:][::-1])), 3)
def a038502(n):
    f=factorint(n)
    return 1 if n==1 else reduce(mul, [1 if i==3 else i**f[i] for i in f])
def a038500(n): return n/a038502(n)
def a263273(n): return 0 if n==0 else a030102(a038502(n))*a038500(n)
def a(n): return a263273(2*n)/2 # Indranil Ghosh, May 23 2017

(define (A263272 n) (/ (A263273 (+ n n)) 2))

a(n) = A263273(2*n) / 2 = A264984(n) / 2.
As a composition of related permutations:
a(n) = A264974(A264975(n)) = A264976(A264974(n)).
Other identities. For all n >= 0:
a(3*n) = 3*a(n).
A000035(a(n)) = A000035(n). [This permutation preserves the parity of n.]
A264974(n) = a(2n)/2. [Thus the restriction onto even numbers induces yet another permutation.]

A060904 Largest power of 5 that divides n.

Original entry on oeis.org

1, 1, 1, 1, 5, 1, 1, 1, 1, 5, 1, 1, 1, 1, 5, 1, 1, 1, 1, 5, 1, 1, 1, 1, 25, 1, 1, 1, 1, 5, 1, 1, 1, 1, 5, 1, 1, 1, 1, 5, 1, 1, 1, 1, 5, 1, 1, 1, 1, 25, 1, 1, 1, 1, 5, 1, 1, 1, 1, 5, 1, 1, 1, 1, 5, 1, 1, 1, 1, 5, 1, 1, 1, 1, 25, 1, 1, 1, 1, 5, 1, 1, 1, 1, 5, 1, 1, 1, 1, 5, 1, 1, 1, 1, 5, 1, 1, 1, 1

Offset: 1

Ahmed Fares (ahmedfares(AT)my-deja.com), May 06 2001

Also the largest power of 5 that divides the n-th Fibonacci number A000045(n).
Multiplicative with a(p^e) = 5^e if p = 5, else a(p^e) = 1. - Mitch Harris, Apr 19 2005
Also 5-adic value of 1/n, n >= 1. See the Mahler reference, definition on p. 7. This is a non-archimedean valuation. See Mahler, p. 10. Sometimes also called 5-adic absolute value. - Wolfdieter Lang, Jun 30 2014

			a(10) = 5 because 10 = 5 * 2.

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

G. C. Greubel, Table of n, a(n) for n = 1..1000
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.

Cf. A000045, A000203, A001620, A038500, A060865, A060901, A112765, A132739, A268354, A268357.

[5^Valuation(n,5): n in [1..100]]; // Vincenzo Librandi, Dec 29 2015

A060904 := n -> 5^padic[ordp](n,5): # Peter Luschny, Nov 26 2010

Table[5^IntegerExponent[n, 5], {n, 100}] (* Vincenzo Librandi, Dec 29 2015 *)

a(n)=5^valuation(n,5) \\ Charles R Greathouse IV, Aug 05 2015

[5^valuation(i,5) for i in [1..100]] # Tom Edgar, Mar 22 2014

If n is not divisible by 5, then a(n) = 1. If n = 5^k * m where m is not divisible by 5, then a(n) = 5^k.
Dirichlet g.f.: zeta(s)*(5^s-1)/(5^s-5). - R. J. Mathar, Jul 12 2012
a(n) = 5^A112765(n). - Tom Edgar, Mar 22 2014
From Peter Bala, Feb 21 2019: (Start)
a(n) = gcd(n,5^n).
a(n) = n/A132739(n).
O.g.f.: x/(1 - x) + 4*Sum_{n >= 1} 5^(n-1)*x^(5^n)/ (1 - x^(5^n)). (End).
a(n) = (1/5)*(sigma(5*n) - sigma(n))/(sigma(5*n) - 5*sigma(n)), where sigma(n) = A000203(n). - Peter Bala, Jun 10 2022
Sum_{k=1..n} a(k) ~ (4/(5*log(5)))*n*log(n) + (3/5 + 4*(gamma-1)/(5*log(5)))*n, where gamma is Euler's constant (A001620). - Amiram Eldar, Nov 15 2022

More terms from Larry Reeves (larryr(AT)acm.org), May 07 2001

Edited by Joerg Arndt and M. F. Hasler, Dec 29 2015

A234957 Highest power of 4 dividing n.

Original entry on oeis.org

1, 1, 1, 4, 1, 1, 1, 4, 1, 1, 1, 4, 1, 1, 1, 16, 1, 1, 1, 4, 1, 1, 1, 4, 1, 1, 1, 4, 1, 1, 1, 16, 1, 1, 1, 4, 1, 1, 1, 4, 1, 1, 1, 4, 1, 1, 1, 16, 1, 1, 1, 4, 1, 1, 1, 4, 1, 1, 1, 4, 1, 1, 1, 64, 1, 1, 1, 4, 1, 1, 1, 4, 1, 1, 1, 4, 1, 1, 1, 16

Offset: 1

Tom Edgar, Jan 01 2014

The generalized binomial coefficients produced by this sequence provide an analog to Kummer's Theorem using arithmetic in base 4.

In the binary representation of n, remove zeros from the right until the number of zeros is even, then remove all but the rightmost one bit. - Ralf Stephan, Jan 05 2014

			Since 8=4*2, then a(8)=4. Likewise, since 4 does not divide 9, a(9)=1.

G. C. Greubel, Table of n, a(n) for n = 1..1000
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.
Index entries for sequences related to binary expansion of n.

Cf. A001620, A006519, A038500.

Table[4^(IntegerExponent[n, 4]), {n, 1, 50}] (* G. C. Greubel, Apr 13 2017 *)

a(n)=4^valuation(n,4) \\ Charles R Greathouse IV, Aug 05 2015

def A234957(n): return 1<<((~n&n-1).bit_length()&-2) # Chai Wah Wu, Jul 08 2022

n=200 #change n for more terms
[4^(valuation(i,4)) for i in [1..n]]

a(n) = 4^(valuation(n,4)).
a(n) = 4^(floor(valuation(n,2)/2)) = 4^A004526(A007814(n)). Recurrence: a(4n) = 4a(n), a(4n+k) = 1 for k=1,2,3. - Ralf Stephan, Jan 05 2014
G.f.: x/(1 - x) + 3 * Sum_{k>=1} 4^(k-1)*x^(4^k)/(1 - x^(4^k)). - Ilya Gutkovskiy, Jul 10 2019
From Amiram Eldar, Dec 31 2022: (Start)
Multiplicative with a(2^e) = 2^(2*floor(e/2)), and a(p^e) = 1 if p >= 3.
Dirichlet g.f.: zeta(s)*(4^s-1)/(4^s-4).
Sum_{k=1..n} a(k) ~ (3/(8*log(2)))*n*log(n) + (5/8 + 3*(gamma-1)/(8*log(2)))*n, where gamma is Euler's constant (A001620). (End)

Keyword:mult added by Andrew Howroyd, Jul 23 2018

A264974 Self-inverse permutation of natural numbers: a(n) = A263273(4*n) / 4.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 16, 9, 10, 19, 12, 13, 14, 15, 8, 17, 18, 11, 20, 21, 34, 43, 48, 25, 52, 27, 28, 55, 30, 37, 46, 57, 22, 49, 36, 31, 58, 39, 40, 41, 42, 23, 50, 45, 32, 59, 24, 35, 44, 51, 26, 53, 54, 29, 56, 33, 38, 47, 60, 61, 142, 63, 88, 169, 102, 115, 124, 129, 70, 151, 144, 97, 178, 75, 106, 133, 156, 79, 160, 81

Offset: 0

Antti Karttunen, Dec 05 2015

Antti Karttunen, Table of n, a(n) for n = 0..4921
Index entries for sequences that are permutations of the natural numbers

Terms of A264986 halved.
Cf. A263272, A263273, A264978.
Cf. also A264975, A264976.

from sympy import factorint
from sympy.ntheory.factor_ import digits
from operator import mul
def a030102(n): return 0 if n==0 else int(''.join(map(str, digits(n, 3)[1:][::-1])), 3)
def a038502(n):
    f=factorint(n)
    return 1 if n==1 else reduce(mul, [1 if i==3 else i**f[i] for i in f])
def a038500(n): return n/a038502(n)
def a263273(n): return 0 if n==0 else a030102(a038502(n))*a038500(n)
def a(n): return a263273(4*n)/4 # Indranil Ghosh, May 25 2017

a(n) = A263273(4*n) / 4.
a(n) = A264986(n) / 2 = A263272(2*n) / 2.
As a composition of related permutations:
a(n) = A264975(A263272(n)) = A263272(A264976(n)).
Other identities. For all n >= 0:
a(3*n) = 3*a(n).
A000035(a(n)) = A000035(n). [This permutation preserves the parity of n.]
A264978(n) = a(2n)/2. [Thus the restriction onto even numbers induces yet another permutation.]

A264985 Self-inverse permutation of nonnegative integers: a(n) = (A264983(n)-1) / 2.

Original entry on oeis.org

0, 1, 3, 2, 4, 9, 6, 10, 12, 5, 7, 11, 8, 13, 27, 18, 28, 36, 15, 19, 33, 24, 31, 30, 21, 37, 39, 14, 16, 32, 23, 22, 29, 20, 34, 38, 17, 25, 35, 26, 40, 81, 54, 82, 108, 45, 55, 99, 72, 85, 90, 63, 109, 117, 42, 46, 96, 69, 58, 87, 60, 100, 114, 51, 73, 105, 78, 94, 84, 57, 91, 111, 48, 64, 102, 75, 112, 93, 66, 118, 120, 41

Offset: 0

Antti Karttunen, Dec 05 2015

nonn

Antti Karttunen, Table of n, a(n) for n = 0..9841
Index entries for sequences that are permutations of the natural numbers

Cf. A263273, A264983.

Cf. also A264989, A264991, A264992.

f[n_] := Block[{g, h}, g[x_] := x/3^IntegerExponent[x, 3]; h[x_] := x/g@ x; If[n == 0, 0, FromDigits[Reverse@ IntegerDigits[#, 3], 3] &@ g[n] h[n]]]; t = Select[f /@ Range@ 1000, OddQ]; Table[(t[[n + 1]] - 1)/2, {n, 0, 81}] (* Michael De Vlieger, Jan 04 2016, after Jean-François Alcover at A263273 *)

from sympy import factorint
from sympy.ntheory.factor_ import digits
from operator import mul
def a030102(n): return 0 if n==0 else int(''.join(map(str, digits(n, 3)[1:][::-1])), 3)
def a038502(n):
    f=factorint(n)
    return 1 if n==1 else reduce(mul, [1 if i==3 else i**f[i] for i in f])
def a038500(n): return n/a038502(n)
def a263273(n): return 0 if n==0 else a030102(a038502(n))*a038500(n)
def a(n): return (a263273(2*n + 1) - 1)/2 # Indranil Ghosh, May 22 2017

(define (A264985 n) (/ (- (A264983 n) 1) 2))

a(n) = (A264983(n)-1) / 2 = (1/2) * (A263273(2n + 1) - 1).

A265352 Permutation of nonnegative integers: a(n) = A263273(A263272(n)).

Original entry on oeis.org

0, 1, 2, 3, 4, 7, 6, 19, 8, 9, 10, 5, 12, 13, 22, 21, 64, 23, 18, 55, 20, 57, 58, 25, 24, 73, 26, 27, 28, 11, 30, 31, 16, 15, 46, 17, 36, 37, 14, 39, 40, 67, 66, 199, 68, 63, 190, 65, 192, 193, 70, 69, 208, 71, 54, 163, 56, 165, 166, 61, 60, 181, 62, 171, 172, 59, 174, 175, 76, 75, 226, 77, 72, 217, 74, 219, 220, 79, 78, 235, 80, 81

Offset: 0

Antti Karttunen, Dec 07 2015

Composition of A263273 with the permutation obtained from its even bisection.

Antti Karttunen, Table of n, a(n) for n = 0..9841
Index entries for sequences that are permutations of the natural numbers

Inverse: A265351.
Cf. A263272, A263273, A264974, A265368.
Cf. also A265354, A265355, A265356

from sympy import factorint
from sympy.ntheory.factor_ import digits
from operator import mul
def a030102(n): return 0 if n==0 else int(''.join(map(str, digits(n, 3)[1:][::-1])), 3)
def a038502(n):
    f=factorint(n)
    return 1 if n==1 else reduce(mul, [1 if i==3 else i**f[i] for i in f])
def a038500(n): return n/a038502(n)
def a263273(n): return 0 if n==0 else a030102(a038502(n))*a038500(n)
def a(n): return a263273(a263273(2*n)/2) # Indranil Ghosh, Jun 08 2017

(define (A265352 n) (A263273 (A263272 n)))

a(n) = A263273(A263272(n)).
As a composition of other related permutations:
a(n) = A265368(A264974(n)).
Other identities. For all n >= 0:
a(3*n) = 3*a(n).

A355582 a(n) is the largest 5-smooth divisor of n.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 1, 8, 9, 10, 1, 12, 1, 2, 15, 16, 1, 18, 1, 20, 3, 2, 1, 24, 25, 2, 27, 4, 1, 30, 1, 32, 3, 2, 5, 36, 1, 2, 3, 40, 1, 6, 1, 4, 45, 2, 1, 48, 1, 50, 3, 4, 1, 54, 5, 8, 3, 2, 1, 60, 1, 2, 9, 64, 5, 6, 1, 4, 3, 10, 1, 72, 1, 2, 75, 4, 1, 6, 1, 80

Offset: 1

Amiram Eldar, Jul 08 2022

Amiram Eldar, Table of n, a(n) for n = 1..10000
Vaclav Kotesovec, Graph - the asymptotic ratio (1000000 terms)

Cf. A006519, A007814, A007949, A038500, A051037, A060904, A065331, A112765, A132741, A165725, A355583, A355584.

Cf. A379005 (rgs-transform), A379006 (ordinal transform).

a[n_] := Times @@ ({2, 3, 5}^IntegerExponent[n, {2, 3, 5}]); Array[a, 100]

a(n) = 3^valuation(n, 3) * 5^valuation(n, 5) << valuation(n, 2);

from sympy import multiplicity as v
def a(n): return 2**v(2, n) * 3**v(3, n) * 5**v(5, n)
print([a(n) for n in range(1, 81)]) # Michael S. Branicky, Jul 08 2022

Multiplicative with a(p^e) = p^e if p <= 5 and 1 otherwise.
a(n) = A006519(n) * A038500(n) * A060904(n).
a(n) = 2^A007814(n) * 3^A007949(n) * 5^A112765(n).
a(n) = n / A165725(n).
Dirichlet g.f.: zeta(s)*(2^s-1)*(3^s-1)*(5^s-1)/((2^s-2)*(3^s-3)*(5^s-5)). - Amiram Eldar, Dec 25 2022
Sum_{k=1..n} a(k) ~ 2*n*log(n)^3 / (45*log(2)*log(3)*log(5)) + O(n*log(n)^2). - Vaclav Kotesovec, Apr 20 2025

A060828 Size of the Sylow 3-subgroup of the symmetric group S_n.

Original entry on oeis.org

1, 1, 1, 3, 3, 3, 9, 9, 9, 81, 81, 81, 243, 243, 243, 729, 729, 729, 6561, 6561, 6561, 19683, 19683, 19683, 59049, 59049, 59049, 1594323, 1594323, 1594323, 4782969, 4782969, 4782969, 14348907, 14348907, 14348907, 129140163, 129140163, 129140163, 387420489

Offset: 0

Ahmed Fares (ahmedfares(AT)my-deja.com), Apr 30 2001

			a(3) = 3 because in S_3 the Sylow 3-subgroup is the subgroup generated by the 3-cycles (123) and (132), its order is 3.

Harry J. Smith, Table of n, a(n) for n = 0..200
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.
Index entries for sequences related to groups

Cf. A054861, A060818.

(* By the formula: *) Table[3^IntegerExponent[n!, 3], {n, 0, 40}] (* Bruno Berselli, Aug 05 2013 *)

for (n=0, 200, s=0; d=3; while (n>=d, s+=n\d; d*=3); write("b060828.txt", n, " ", 3^s)) \\ Harry J. Smith, Jul 12 2009

def A060828(n):
    A004128 = lambda n: A004128(n//3) + n if n > 0 else 0
    return 3^A004128(n//3)
[A060828(i) for i in (0..39)]  # Peter Luschny, Nov 16 2012

a(n) = 3^A054861(n) = 3^(floor(n/3) + floor(n/9) + floor(n/27) + floor(n/81) + ...).
a(n) = Product_{i=1..n} A038500(i). - Tom Edgar, Apr 30 2014
a(n) = 3^(n/2 + O(log n)). - Charles R Greathouse IV, Aug 05 2015

More terms from N. J. A. Sloane, Jul 03 2008

A265351 Permutation of nonnegative integers: a(n) = A263272(A263273(n)).

Original entry on oeis.org

0, 1, 2, 3, 4, 11, 6, 5, 8, 9, 10, 29, 12, 13, 38, 33, 32, 35, 18, 7, 20, 15, 14, 17, 24, 23, 26, 27, 28, 83, 30, 31, 92, 87, 86, 89, 36, 37, 110, 39, 40, 119, 114, 113, 116, 99, 34, 101, 96, 95, 98, 105, 104, 107, 54, 19, 56, 21, 22, 65, 60, 59, 62, 45, 16, 47, 42, 41, 44, 51, 50, 53, 72, 25, 74, 69, 68, 71, 78, 77, 80, 81

Offset: 0

Antti Karttunen, Dec 07 2015

Composition of A263273 with the permutation obtained from its even bisection.

Antti Karttunen, Table of n, a(n) for n = 0..9841
Index entries for sequences that are permutations of the natural numbers

Inverse: A265352.
Cf. A263272, A263273, A264974, A265367.
Cf. also A265342, A265353, A265355, A265356.

from sympy import factorint
from sympy.ntheory.factor_ import digits
from operator import mul
def a030102(n): return 0 if n==0 else int(''.join(map(str, digits(n, 3)[1:][::-1])), 3)
def a038502(n):
    f=factorint(n)
    return 1 if n==1 else reduce(mul, [1 if i==3 else i**f[i] for i in f])
def a038500(n): return n/a038502(n)
def a263273(n): return 0 if n==0 else a030102(a038502(n))*a038500(n)
def a263272(n): return a263273(2*n)/2
def a(n): return a263272(a263273(n)) # Indranil Ghosh, May 25 2017

(define (A265351 n) (A263272 (A263273 n)))

a(n) = A263272(A263273(n)).
As a composition of other related permutations:
a(n) = A264974(A265367(n)).
Other identities. For all n >= 0:
a(3*n) = 3*a(n).
a(n) = A265342(n)/2.

A265354 Permutation of nonnegative integers: a(n) = A263273(A264985(n)).

Original entry on oeis.org

0, 1, 3, 2, 4, 9, 6, 10, 12, 7, 5, 19, 8, 13, 27, 18, 28, 36, 21, 11, 57, 24, 37, 30, 15, 31, 39, 22, 16, 64, 23, 14, 55, 20, 46, 58, 25, 17, 73, 26, 40, 81, 54, 82, 108, 63, 29, 171, 72, 109, 90, 45, 85, 117, 66, 34, 192, 69, 38, 165, 60, 100, 174, 75, 35, 219, 78, 118, 84, 33, 91, 93, 48, 32, 138, 51, 112, 111, 42, 94, 120, 67

Offset: 0

Antti Karttunen, Dec 07 2015

Composition of A263273 with the permutation obtained from its odd bisection.

Antti Karttunen, Table of n, a(n) for n = 0..9841
Index entries for sequences that are permutations of the natural numbers

Inverse: A265353.
Cf. A263273, A264985.
Cf. also A265352, A265355, A265356.

from sympy import factorint
from sympy.ntheory.factor_ import digits
from operator import mul
def a030102(n): return 0 if n==0 else int(''.join(map(str, digits(n, 3)[1:][::-1])), 3)
def a038502(n):
    f=factorint(n)
    return 1 if n==1 else reduce(mul, [1 if i==3 else i**f[i] for i in f])
def a038500(n): return n/a038502(n)
def a263273(n): return 0 if n==0 else a030102(a038502(n))*a038500(n)
def a264985(n): return (a263273(2*n + 1) - 1)/2
def a(n): return a263273(a264985(n)) # Indranil Ghosh, May 22 2017

(define (A265354 n) (A263273 (A264985 n)))

a(n) = A263273(A264985(n)).

cp's OEIS Frontend

A263272 Self-inverse permutation of nonnegative integers: a(n) = A263273(2*n) / 2.

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A060904 Largest power of 5 that divides n.

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A234957 Highest power of 4 dividing n.

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A264974 Self-inverse permutation of natural numbers: a(n) = A263273(4*n) / 4.

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A264985 Self-inverse permutation of nonnegative integers: a(n) = (A264983(n)-1) / 2.

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A265352 Permutation of nonnegative integers: a(n) = A263273(A263272(n)).

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A355582 a(n) is the largest 5-smooth divisor of n.

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