A054861 -id:A054861 - cp's OEIS frontend

A004128 a(n) = Sum_{k=1..n} floor(3*n/3^k).

0, 1, 2, 4, 5, 6, 8, 9, 10, 13, 14, 15, 17, 18, 19, 21, 22, 23, 26, 27, 28, 30, 31, 32, 34, 35, 36, 40, 41, 42, 44, 45, 46, 48, 49, 50, 53, 54, 55, 57, 58, 59, 61, 62, 63, 66, 67, 68, 70, 71, 72, 74, 75, 76, 80, 81, 82, 84, 85, 86, 88, 89, 90, 93, 94, 95, 97, 98, 99, 101, 102

Offset: 0

N. J. A. Sloane

nonn

3-adic valuation of (3n)!; cf. A054861.

Denominators of expansion of (1-x)^{-1/3} are 3^a(n). Numerators are in |A067622|.

Gary W. Adamson, in "Beyond Measure, A Guided Tour Through Nature, Myth and Number", by Jay Kappraff, World Scientific, 2002, p. 356.

T. D. Noe, Table of n, a(n) for n = 0..1000
Vaclav Kotesovec, Graph - the asymptotic ratio (100000 terms)

Cf. A004117, A001511, A051064, A055457.

Cf. A054861, A067080, A098844, A132027, A005187, A054899.

a004128 n = a004128_list !! (n-1)
a004128_list = scanl (+) 0 a051064_list
-- Reinhard Zumkeller, May 23 2013

[n + Valuation(Factorial(n), 3): n in [0..70]]; // Vincenzo Librandi, Jun 12 2019

A004128 := proc(n)
    A054861(3*n) ;
end proc:
seq(A004128(n),n=0..100) ; # R. J. Mathar, Nov 04 2017

Table[Total[NestWhileList[Floor[#/3] &, n, # > 0 &]], {n, 0, 70}] (* Birkas Gyorgy, May 20 2012 *)
A004128 = Log[3, CoefficientList[ Series[1/(1+x)^(1/3), {x, 0, 100}], x] // Denominator] (* Jean-François Alcover, Feb 19 2015 *)
Flatten[{0, Accumulate[Table[IntegerExponent[3*n, 3], {n, 1, 100}]]}] (* Vaclav Kotesovec, Oct 17 2019 *)

{a(n) = my(s, t=1); while(t<=n, s += n\t; t*=3);s}; /* Michael Somos, Feb 26 2004 */

a(n) = (3*n-sumdigits(n,3))/2; \\ Christian Krause, Jun 10 2025

def A007949(n):
    c = 0
    while not (a:=divmod(n,3))[1]:
        c += 1
        n = a[0]
    return c
def A004128(n): return n+sum(A007949(i) for i in range(3,n+1)) # Chai Wah Wu, Feb 28 2025

A004128 = lambda n: A004128(n//3) + n if n > 0 else 0
[A004128(n) for n in (0..70)]  # Peter Luschny, Nov 16 2012

A051064(n) = a(n+1) - a(n). - Alford Arnold, Jul 19 2000
a(n) = n + floor(n/3) + floor(n/9) + floor(n/27) + ... = n + a(floor(n/3)) = n + A054861(n) = A054861(3n) = (3*n - A053735(n))/2. - Henry Bottomley, May 01 2001
a(n) = Sum_{k>=0} floor(n/3^k). a(n) = Sum_{k=0..floor(log_3(n))} floor(n/3^k), n >= 1. - Hieronymus Fischer, Aug 14 2007
Recurrence: a(n) = n + a(floor(n/3)); a(3n) = 3*n + a(n); a(n*3^m) = 3*n*(3^m-1)/2 + a(n). - Hieronymus Fischer, Aug 14 2007
a(k*3^m) = k*(3^(m+1)-1)/2, 0 <= k < 3, m >= 0. - Hieronymus Fischer, Aug 14 2007
Asymptotic behavior: a(n) = (3/2)*n + O(log(n)), a(n+1) - a(n) = O(log(n)); this follows from the inequalities below. - Hieronymus Fischer, Aug 14 2007
a(n) <= (3n-1)/2; equality holds for powers of 3. - Hieronymus Fischer, Aug 14 2007
a(n) >= (3n-2)/2 - floor(log_3(n)); equality holds for n = 3^m - 1, m > 0. - Hieronymus Fischer, Aug 14 2007
Lim inf (3n/2 - a(n)) = 1/2, for n->oo. - Hieronymus Fischer, Aug 14 2007
Lim sup (3n/2 - log_3(n) - a(n)) = 0, for n->oo. - Hieronymus Fischer, Aug 14 2007
Lim sup (a(n+1) - a(n) - log_3(n)) = 1, for n->oo. - Hieronymus Fischer, Aug 14 2007
G.f.: (Sum_{k>=0} x^(3^k)/(1-x^(3^k)))/(1-x). - Hieronymus Fischer, Aug 14 2007
a(n) = Sum_{k>=0} A030341(n,k)*A003462(k+1). - Philippe Deléham, Oct 21 2011
a(n) ~ 3*n/2 - log(n)/(2*log(3)). - Vaclav Kotesovec, Oct 17 2019

Current definition suggested by Jason Earls, Jul 04 2001

A090622 Square array read by antidiagonals of highest power of k dividing n! (with n,k>1).

Original entry on oeis.org

1, 0, 1, 0, 1, 3, 0, 0, 1, 3, 0, 0, 1, 1, 4, 0, 1, 0, 1, 2, 4, 0, 0, 1, 1, 2, 2, 7, 0, 0, 0, 1, 1, 2, 2, 7, 0, 0, 1, 0, 2, 1, 3, 4, 8, 0, 0, 0, 1, 0, 2, 1, 3, 4, 8, 0, 0, 0, 0, 1, 1, 2, 1, 4, 4, 10, 0, 0, 0, 1, 1, 1, 1, 4, 2, 4, 5, 10, 0, 0, 1, 0, 1, 1, 2, 1, 4, 2, 5, 5, 11, 0, 0, 0, 1, 0, 1, 1, 2, 1, 4, 2, 5, 5, 11

Offset: 2

Henry Bottomley, Dec 06 2003

			Square array starts:
1, 0, 0, 0, 0, 0, 0, ...
1, 1, 0, 0, 1, 0, 0, ...
3, 1, 1, 0, 1, 0, 1, ...
3, 1, 1, 1, 1, 0, 1, ...
4, 2, 2, 1, 2, 0, 1, ...
4, 2, 2, 1, 2, 1, 1, ...
7, 2, 3, 1, 2, 1, 2, ...

Alois P. Heinz, Antidiagonals n = 2..142, flattened

Columns include A011371, A054861, A090616, A027868, A054861, A054896, A090617, A090618, A027868, A064458, A090619, A090620, A054896, A027868, A090621.
Cf. A090623, A090624, A115627.
Diagonal gives A011776. - Alois P. Heinz, Oct 04 2012

f:= proc(n, p) local c, k; c, k:= 0, p;
       while n>=k do c:= c+iquo(n, k); k:= k*p od; c
    end:
T:= (n, k)-> min(seq(iquo(f(n, i[1]), i[2]), i=ifactors(k)[2])):
seq(seq(T(n, 2+d-n), n=2..d), d=2..20);  # Alois P. Heinz, Oct 04 2012

f[n_, p_] := Module[{c = 0, k = p}, While[n >= k , c = c + Quotient[n, k]; k = k*p ]; c ]; t[n_, k_] := Min[ Table[ Quotient[f[n, i[[1]]], i[[2]]], {i, FactorInteger[k]}]]; Table[ Table[t[n, 2 + d - n], {n, 2, d}], {d, 2, 20}] // Flatten (* Jean-François Alcover, Oct 03 2013, translated from Alois P. Heinz's Maple program *)

For k=p prime: T(n,p) = [n/p] + [n/p^2] + [n/p^3] + .... For k = p^m a prime power: T(n,p^m) = [T(n,p)/m]. For k = b*c with b and c coprime: T(n,a*b) = min(T(n,a), T(n,b)). T(n,k) is close to, but below, n/A090624(k).

A054896 a(n) = Sum_{k>0} floor(n/7^k).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 6, 8, 8, 8, 8, 8, 8, 8, 9, 9, 9, 9, 9, 9, 9, 10, 10, 10, 10, 10, 10, 10, 11, 11, 11, 11, 11, 11, 11, 12, 12, 12, 12, 12, 12, 12, 13, 13, 13, 13, 13, 13

Offset: 0

Henry Bottomley, May 23 2000

nonn

Exponent of the highest power of 7 dividing n!.

			  a(10^0) = 0.
  a(10^1) = 1.
  a(10^3) = 16.
  a(10^3) = 164.
  a(10^4) = 1665.
  a(10^5) = 16662.
  a(10^6) = 166664.
  a(10^7) = 1666661.
  a(10^8) = 16666662.
  a(10^9) = 166666661

Hieronymus Fischer, Table of n, a(n) for n = 0..10000

Cf. A011371 and A054861 for analogs involving powers of 2 and 3.
Cf. A053828, A054893, A054895, A054899, A067080, A098844, A132031, A214411.
Cf. A000142, A214411.

function A054896(n)
  if n eq 0 then return n;
  else return A054896(Floor(n/7)) + Floor(n/7);
  end if; return A054896;
end function;
[A054896(n): n in [0..100]]; // G. C. Greubel, Feb 09 2023

Table[t=0; p=7; While[s=Floor[n/p]; t=t+s; s>0, p *= 7]; t, {n,0,100}]

a(n)=(n-sumdigits(n, 7))\6 \\ Alan Michael Gómez Calderón, Oct 08 2024

def A054896(n):
    if (n==0): return 0
    else: return A054896(n//7) + (n//7)
[A054896(n) for n in range(101)] # G. C. Greubel, Feb 09 2023

a(n) = floor(n/7) + floor(n/49) + floor(n/343) + floor(n/2401) + ...
a(n) = (n - A053828(n))/6.
From Hieronymus Fischer, Aug 14 2007: (Start)
a(n) = a(floor(n/7)) + floor(n/7).
a(7*n) = n + a(n).
a(n*7^m) = a(n) + n*(7^m-1)/6.
a(k*7^m) = k*(7^m-1)/6, for 0 <= k < 7, m >= 0.
Asymptotic behavior:
a(n) = n/6 + O(log(n)).
a(n+1) - a(n) = O(log(n)); this follows from the inequalities below.
a(n) <= (n-1)/6; equality holds for powers of 7.
a(n) >= (n-6)/6 - floor(log_7(n)); equality holds for n=7^m-1, m>0. -
lim inf (n/6 - a(n)) = 1/6, for n-->oo.
lim sup (n/6 - log_7(n) - a(n)) = 0, for n-->oo.
lim sup (a(n+1) - a(n) - log_7(n)) = 0, for n-->oo.
G.f.: (1/(1-x))*Sum_{k > 0} x^(7^k)/(1-x^(7^k)). (End)
Partial sums of A214411. - R. J. Mathar, Jul 08 2021
a(n) = A214411(A000142(n)). - Michel Marcus, Oct 07 2024

Examples added by Hieronymus Fischer, Jun 06 2012

A054895 a(n) = Sum_{k>0} floor(n/6^k).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 7, 7, 7, 7, 7, 7, 8, 8, 8, 8, 8, 8, 9, 9, 9, 9, 9, 9, 10, 10, 10, 10, 10, 10, 11, 11, 11, 11, 11, 11, 12, 12, 12, 12, 12, 12, 14, 14, 14, 14, 14, 14, 15, 15, 15, 15, 15, 15, 16, 16, 16

Offset: 0

Henry Bottomley, May 23 2000

nonn

Different from the highest power of 6 dividing n! (cf. A054861). - Hieronymus Fischer, Aug 14 2007

Partial sums of A122841. - Hieronymus Fischer, Jun 06 2012

			  a(10^0) = 0.
  a(10^1) = 1.
  a(10^2) = 18.
  a(10^3) = 197.
  a(10^4) = 1997.
  a(10^5) = 19996.
  a(10^6) = 199995.
  a(10^7) = 1999995.
  a(10^8) = 19999994.
  a(10^9) = 199999993.

Hieronymus Fischer, Table of n, a(n) for n = 0..10000

Cf. A011371 and A054861 for analogs involving powers of 2 and 3.

Cf. A053827, A054861, A054899, A067080, A098844, A122841, A132030.

a054895 n = a054895_list !! n
a054895_list = scanl (+) 0 a122841_list
-- Reinhard Zumkeller, Nov 10 2013

function A054895(n)
  if n eq 0 then return n;
  else return A054895(Floor(n/6)) + Floor(n/6);
  end if; return A054895;
end function;
[A054895(n): n in [0..100]]; // G. C. Greubel, Feb 09 2023

Table[t=0; p=6; While[s=Floor[n/p]; t=t+s; s>0, p *= 6]; t, {n,0,100}]

def A054895(n):
    if (n==0): return 0
    else: return A054895(n//6) + (n//6)
[A054895(n) for n in range(104)] # G. C. Greubel, Feb 09 2023

a(n) = floor(n/6) + floor(n/36) + floor(n/216) + floor(n/1296) + ...
a(n) = (n - A053827(n))/5.
From Hieronymus Fischer, Aug 14 2007: (Start)
a(n) = a(floor(n/6)) + floor(n/6).
a(6*n) = n + a(n).
a(n*6^m) = n*(6^m-1)/5 + a(n).
a(k*6^m) = k*(6^m-1)/5, for 0 <= k < 6, m >= 0.
Asymptotic behavior:
a(n) = (n/5) + O(log(n)).
a(n+1) - a(n) = O(log(n)); this follows from the inequalities below.
a(n) <= (n-1)/5; equality holds for powers of 6.
a(n) >= ((n-5)/5) - floor(log_6(n)); equality holds for n=6^m-1, m>0.
lim inf (n/5 - a(n)) = 1/5, for n-->oo.
lim sup (n/5 - log_6(n) - a(n)) = 0, for n-->oo.
lim sup (a(n+1) - a(n) - log_6(n)) = 0, for n-->oo.
G.f.: (1/(1-x))*Sum_{k > 0} x^(6^k)/(1-x^(6^k)). (End)

An incorrect formula was deleted by N. J. A. Sloane, Nov 18 2008

Examples added by Hieronymus Fischer, Jun 06 2012

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

A261234 a(n) = number of steps to reach (3^n)-1 when starting from k = (3^(n+1))-1 and repeatedly applying the map that replaces k with k - (sum of digits in base-3 representation of k).

Original entry on oeis.org

1, 2, 5, 12, 29, 74, 196, 530, 1445, 3956, 10862, 29901, 82592, 229233, 639967, 1797288, 5073707, 14381347, 40890492, 116559600, 333043360, 953890490, 2738788806, 7881915828, 22729464587, 65652788211, 189866467219, 549596773550, 1592118137130, 4615680732717, 13392399641613, 38894563977633, 113074467549440, 329080350818600, 958725278344368, 2795854777347489

Offset: 0

Antti Karttunen, Aug 13 2015

Hiroaki Yamanouchi, Table of n, a(n) for n = 0..100
Antti Karttunen, Naive C-program for computing the terms of A261234, A261236 and A261237 at the same time
Hiroaki Yamanouchi, Fast Python-program for computing terms of A213709, A261234 and analogous sequences in other bases

Cf. A000244, A054861, A261231, A261230.
First differences of A261232 and A261233.
Sum of A261236 and A261237.
Cf. A261235 (first differences of this sequence).
Cf. also A213709.

Table[Length@ NestWhileList[# - Total@ IntegerDigits[#, 3] &, 3^(n + 1) - 1, # > 3^n - 1 &] - 1, {n, 0, 16}] (* Michael De Vlieger, Jun 27 2016 *)

a(n) = A261236(n) + A261237(n).

a(23)-a(35) from Hiroaki Yamanouchi, Aug 16 2015

A276623 The infinite trunk of ternary beanstalk: The only infinite sequence such that a(n-1) = a(n) - A053735(a(n)), where A053735(n) = base-3 digit sum of n.

Original entry on oeis.org

0, 2, 4, 8, 10, 12, 16, 20, 26, 28, 30, 34, 38, 42, 46, 52, 56, 62, 68, 72, 80, 82, 84, 88, 92, 96, 100, 106, 110, 116, 122, 126, 134, 140, 144, 152, 160, 164, 170, 176, 180, 188, 194, 198, 204, 212, 216, 224, 232, 242, 244, 246, 250, 254, 258, 262, 268, 272, 278, 284, 288, 296, 302, 306, 314, 322, 326, 332, 338, 342, 350, 356, 360

Offset: 0

Antti Karttunen, Sep 11 2016

Antti Karttunen, Table of n, a(n) for n = 0..6250

Cf. A004128, A024023, A053735, A054861, A261231 (left inverse), A261233, A276622, A276624, A276603 (terms divided by 2), A276604 (first differences).
Cf. A179016, A219648, A219666, A255056, A259934, A276573, A276583, A276613 for similar constructions.
Cf. also A263273.

(define (A276623 n) (A276624 (A276622 n)))

a(n) = A276624(A276622(n)).
Other identities. For all n >= 0:
A261231(a(n)) = n.
a(A261233(n)) = A024023(n) = 3^n - 1.

A090616 Exponent of highest power of 4 dividing n!.

Original entry on oeis.org

0, 0, 0, 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 5, 5, 7, 7, 8, 8, 9, 9, 9, 9, 11, 11, 11, 11, 12, 12, 13, 13, 15, 15, 16, 16, 17, 17, 17, 17, 19, 19, 19, 19, 20, 20, 21, 21, 23, 23, 23, 23, 24, 24, 25, 25, 26, 26, 27, 27, 28, 28, 28, 28, 31, 31, 32, 32, 33, 33, 33, 33, 35, 35, 35, 35, 36

Offset: 0

Henry Bottomley, Dec 06 2003

nonn

			a(6)=2 since 6! = 720 = 4^2 * 45.

Vincenzo Librandi, Table of n, a(n) for n = 0..10000

Cf. A011371, A054861, A027868, A054896, A235127.

IntegerExponent[Range[0, 100]!, 4] (* Vincenzo Librandi, Mar 10 2013 *)

a(n) = valuation( n!, 4 ); /* Joerg Arndt, Mar 10 2013 */
(Python 3.10+)
def A090616(n): return (n-n.bit_count())>>1 # Chai Wah Wu, Jul 09 2022

a(n) = A090622(n, 4) = floor(A011371(n)/2) = floor((floor(n/2) + floor(n/4) + floor(n/8) + floor(n/16) + ...)/2).

a(n) = A235127(n!). - R. J. Mathar, Jul 08 2021

A268395 Partial sums of A268389.

Original entry on oeis.org

0, 0, 0, 1, 1, 3, 4, 4, 4, 5, 7, 7, 8, 8, 8, 11, 11, 15, 16, 16, 18, 18, 18, 19, 20, 20, 20, 22, 22, 23, 26, 26, 26, 27, 31, 31, 32, 32, 32, 34, 36, 36, 36, 37, 37, 40, 41, 41, 42, 42, 42, 47, 47, 48, 50, 50, 50, 52, 53, 53, 56, 56, 56, 57, 57, 59, 60, 60, 64, 64, 64, 65, 66, 66, 66, 69, 69, 70, 72, 72, 74, 74, 74, 75, 75, 81

Offset: 0

Antti Karttunen, Feb 10 2016

nonn

Antti Karttunen, Table of n, a(n) for n = 0..65537

Cf. A048631, A268389, A268672.
Cf. A268678 (with duplicates removed), A268677 (numbers that do not occur here).
Cf. A268708, A268709, A268710.
Cf. also A054861.

f[n_] := Which[n == 1, 0, OddQ@ #, 0, EvenQ@ #, 1 + f[#/2]] &@ Fold[BitXor, n, Quotient[n, 2^Range[BitLength@ n - 1]]]; Accumulate@ Array[f, {85}] (* Michael De Vlieger, Feb 12 2016, after Jan Mangaldan at A006068 *)

a(0) = 0, for n >= 1, a(n) = A268389(n) + a(n-1).
Other identities. For all n >= 0:
a(n) = A268389(A048631(n)).
a(n) = n - A268672(n).

A004117 Numerators of expansion of (1-x)^(-1/3).

Original entry on oeis.org

1, 1, 2, 14, 35, 91, 728, 1976, 5434, 135850, 380380, 1071980, 9111830, 25933670, 74096200, 637227320, 1832028545, 5280552865, 137294374490, 397431084050, 1152550143745, 10043651252635, 29217894553120, 85112997176480

Offset: 0

N. J. A. Sloane

nonn

For n >= 1, a(n) is also the numerator of beta(n+1/3,2/3)*sqrt(27)/(2*Pi). - Groux Roland, May 17 2011

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Paveł Szabłowski, Beta distributions whose moment sequences are related to integer sequences listed in the OEIS, Contrib. Disc. Math. (2024) Vol. 19, No. 4, 85-109. See p. 93.

Cf. A004128.

Cf. A004987, A007559, A047657, A054861, A034689, A053101, A072888.

Table[Numerator[Binomial[-1/3,n] (-1)^n],{n,0,40}] (* Vincenzo Librandi, Jun 13 2012 *)

a(n)=prod(k=1,n,3*k-2)/n!*3^sum(k=1,n,valuation(k,3))

(1/n!) * 3^A054861(n) * Product_{k=0..n-1} (3k+1). - Ralf Stephan, Mar 13 2004

Numerators in (1-3t)^(-1/3) = 1 + t + 2*t^2 + (14/3)*t^3 + (35/3)*t^4 + (91/3)*t^5 + (728/9)*t^6 + (1976/9)*t^7 + (5434/9)*t^8 + ... = 1 + t + 4*t^2/2! + 28*t^3/3! + 280*t^4/4! + 3640*t^5/5! + 58240*t^6/6! + ... = e.g.f. for triple factorials A007559 (cf. A094638). - Tom Copeland, Dec 04 2013

Typo in formula fixed by Pontus von Brömssen, Nov 25 2008

cp's OEIS Frontend

A004128 a(n) = Sum_{k=1..n} floor(3*n/3^k).

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A090622 Square array read by antidiagonals of highest power of k dividing n! (with n,k>1).

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Maple

Mathematica

Formula

A054896 a(n) = Sum_{k>0} floor(n/7^k).

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Magma

Mathematica

PARI

SageMath

Formula

Extensions

A054895 a(n) = Sum_{k>0} floor(n/6^k).

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Programs

Haskell

Magma

Mathematica

SageMath

Formula

Extensions

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

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Programs

Mathematica

PARI

Sage

Formula

Extensions

A261234 a(n) = number of steps to reach (3^n)-1 when starting from k = (3^(n+1))-1 and repeatedly applying the map that replaces k with k - (sum of digits in base-3 representation of k).

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