A060828 -id:A060828 - cp's OEIS frontend

A242849 Triangle read by rows: T(n,k) = A060828(n)/(A060828(k) * A060828(n-k)).

1, 1, 1, 1, 1, 1, 1, 3, 3, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 3, 1, 3, 3, 1, 1, 1, 3, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 9, 9, 3, 9, 9, 3, 9, 9, 1, 1, 1, 9, 3, 3, 9, 3, 3, 9, 1, 1, 1, 1, 1, 3, 3, 3, 3, 3, 3, 1, 1, 1, 1, 3, 3, 1, 9, 9, 3, 9, 9, 1

Offset: 0

Tom Edgar, May 23 2014

This triangle can be obtained by replacing each entry of Pascal's Triangle by the largest power of 3 dividing that entry.
The exponent of T(n,k) is the number of 'carries' that occur when adding k and n-k in base 3 using the traditional addition algorithm.
If T(n,k) != 0 mod 3, then n dominates k in base 3.

			The triangle begins
1
1 1
1 1 1
1 3 3 1
1 1 3 1 1
1 1 1 1 1 1
1 3 3 1 3 3 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.
E. Burlachenko, Fractal generalized Pascal matrices, arXiv:1612.00970 [math.NT], 2016. See p. 7.
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. A007318, A038500, A060828, A082907.

s3[n_] := 3^IntegerExponent[n!, 3];
T[n_, k_] := s3[n]/(s3[k] s3[n-k]);
Table[T[n, k], {n, 0, 12}, {k, 0, n}] // Flatten (* Jean-François Alcover, Dec 06 2018 *)

m=50
T=[0]+[3^valuation(i,3) 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) = A060828(n)/(A060828(k) * A060828(n-k)).
T(n,k) = Product_{i=1..n} A038500(i)/(Product_{i=1..k} A038500(i)*Product_{i=1..n-k} A038500(i)).
T(n,k) = A038500(n)/n*(k/A038500(k)*T(n-1,k-1)+(n-k)/A038500(n-k)*T(n-1,k)).

A136690 Final nonzero digit of n! in base 3.

Original entry on oeis.org

1, 1, 2, 2, 2, 1, 2, 2, 1, 1, 1, 2, 2, 2, 1, 2, 2, 1, 2, 2, 1, 1, 1, 2, 1, 1, 2, 2, 2, 1, 1, 1, 2, 1, 1, 2, 2, 2, 1, 1, 1, 2, 1, 1, 2, 1, 1, 2, 2, 2, 1, 2, 2, 1, 2, 2, 1, 1, 1, 2, 1, 1, 2, 2, 2, 1, 1, 1, 2, 1, 1, 2, 1, 1, 2, 2, 2, 1, 2, 2, 1, 1, 1, 2, 2, 2, 1, 2, 2, 1, 1, 1, 2, 2, 2, 1, 2, 2, 1, 2, 2, 1, 1, 1, 2

Offset: 0

Carl R. White, Jan 16 2008

			6! = 720 decimal = 222200 ternary, so a(6) = 2.

Cf. A000142, A127110.

Other bases: A136691, A136692, A136693, A136694, A136695, A136696, A008904, A136697, A136698, A136699, A136700, A136701, A136702, A212307.

f[n_] := Mod[6 Times @@ (Rest[ FoldList[{1 + #1[[1]], #2! 2^(#1[[1]] #2)} &, {0, 0}, Reverse[ IntegerDigits[n, 3]]]]), 10][[2]]; # /. {0 -> 1} & /@ Mod[Table[f@n, {n, 0, 104}], 3] (* Robert G. Wilson v, Apr 17 2010 *)
fnzd[n_]:=Module[{sidn3=Split[IntegerDigits[n!,3]]},If[MemberQ[ Last[ sidn3],0], sidn3[[-2,1]], sidn3[[-1,1]]]]; Array[fnzd,110,0] (* Harvey P. Dale, May 03 2018 *)

a(n) = vecsum([bittest(220,b) |b<-digits(n,9)])%2 + 1; \\ Kevin Ryde, Dec 03 2022

From David Radcliffe, Sep 03 2021: (Start)
a(n) = (n! / A060828(n)) mod 3;
a(n) = 1 + (A189672(n) mod 2);
a(6*n) = a(6*n+1) = a(2*n);
a(6*n+2) = 3 - a(2*n);
a(6*n+3) = a(6*n+4) = 3 - a(2*n+1);
a(6*n+5) = a(2*n+1).
(End)
a(n) = A008904(A127110(n)). - Michel Marcus, Sep 04 2021
From Kevin Ryde, Dec 03 2022: (Start)
a(n) = 1 if n written in base 9 has an even number of digits {2,3,4,6,7}; and otherwise a(n) = 2.
Fixed point of the morphism 1 -> 1,1,2,2,2,1,2,2,1; 2 -> 2,2,1,1,1,2,1,1,2; starting from 1.
(End)
a(n) = A212307(n) mod 3. - Ridouane Oudra, Sep 25 2024

More terms from Robert G. Wilson v, Apr 17 2010

A090630 Greatest divisor d of n! such that d=m^k with k>1.

Original entry on oeis.org

1, 1, 1, 1, 8, 8, 144, 144, 576, 5184, 518400, 518400, 2073600, 2073600, 101606400, 914457600, 14631321600, 14631321600, 526727577600, 526727577600, 52672757760000, 221225582592000, 6373403688960000, 6373403688960000, 917770131210240000, 22944253280256000000, 3877578804363264000000

Offset: 0

Reinhard Zumkeller, Dec 13 2003

nonn

a(n) is a square for all n except n = 4, 5 and 21 (Wilke, 1981). - Amiram Eldar, Jun 09 2022

Charles R Greathouse IV, Table of n, a(n) for n = 0..500
Eric Weisstein's World of Mathematics, Perfect Powers.
Eric Weisstein's World of Mathematics, Factorial.
Kenneth M. Wilke, Problem 493, Pi Mu Epsilon Journal, Vol. 7, No. 4 (1981), p. 265; alternative link.
Index entries for sequences related to factorial numbers.

Cf. A011776, A060818, A060828, A251753.

f:= proc(n)
  local F,  k, d,r,s;
  F:= ifactors(n!)[2];
  r:= 1;
  for k from 2 to F[1][2] do
    r:= max(r, mul(f[1]^(k*floor(f[2]/k)),f=F))
  od:
r
end proc:
1,1,seq(f(n), n=2..100); # Robert Israel, Dec 08 2014

IsPower[n_] := If[n==1, True, GCD@@(Transpose[FactorInteger[n]][[2]])>1]; Table[Select[Divisors[n! ], IsPower][[ -1]], {n, 0, 25}]

a(n)=my(f=factor(n!),m=1); for(i=2,if(#f~,f[1,2]), m=max(factorback(concat(Mat(f[,1]), f[,2]\i*i)),m)); m \\ Charles R Greathouse IV, Dec 09 2014

a(n)= n!/A251753(n). - Robert G. Wilson v, Dec 08 2014

More terms from T. D. Noe, Oct 04 2004

A232097 a(n) = least k such that 1+2+3+...+k (k-th triangular number) is a multiple of n!; a(n) = least k such that A232096(k) >= n.

Original entry on oeis.org

1, 3, 3, 15, 15, 224, 224, 4095, 76544, 512000, 9511424, 20916224, 410572799, 672358400, 2985984000, 1004293914624, 1004293914624, 78942076928000, 610877575397375, 83179139563520000, 490473044848410624, 6878928869130239999, 185974097225789210624, 1708887984313466880000, 68817755280574852890624

Offset: 1

Antti Karttunen, Nov 18 2013

nonn

a(n) = least k such that A232096(k) >= n.
Each A000217(a(n)) is divisible by A118381(n).
Each a(n) or a(n)+1 is divisible by 2*A060818(n) = A086117(n+1).
Each a(n) or a(n)+1 is divisible by A060828(n), and similarly for all the higher bases.
If we were instead searching for the first occurrence where A232096 gets a new distinct value, then we would have another sequence, b, which would start as: 1, 3, 4, 15, 32, 224, 575, 4095, ... as those distinct values do not appear in monotone order, being for n>=1, A232096(b(n)) = 1, 3, 2, 5, 4, 7, 6, 8, 9, 10, ...

			a(5) = 15 as binomial(15 + 1, 2) = 120 is the smallest binomial that is divisible by 5! = 120. - _David A. Corneth_, Mar 29 2021

Cf. A000217, A232096. A232101 gives the ratio A000217(a(n)) / n!

a(n) = { my(p = 2*n!, f = factor(p), res = oo); for(i = 2^(#f~-1), 2^#f~-1, b = binary(i); pr = prod(j = 1, #f~, f[j,1]^(b[j]*f[j, 2])); ipr = p/pr; for(j = -1, 0, c = lift(chinese(Mod(-1-j, ipr), Mod(j, pr))); if(c > 0, res = min(res, c)))); res } \\ David A. Corneth, Mar 29 2021

(define (A232097 n) (let ((increment (* 2 (A060818 n)))) (let loop ((k increment)) (cond ((>= (A232096 (- k 1)) n) (- k 1)) ((>= (A232096 k) n) k) (else (loop (+ k increment)))))))
;; Alternative, very naive and slow version:
(define (A232097v2 n) (let loop ((k 1)) (if (>= (A232096 k) n) k (loop (+ 1 k)))))

A242954 a(n) = Product_{i=1..n} A234957(i).

Original entry on oeis.org

1, 1, 1, 1, 4, 4, 4, 4, 16, 16, 16, 16, 64, 64, 64, 64, 1024, 1024, 1024, 1024, 4096, 4096, 4096, 4096, 16384, 16384, 16384, 16384, 65536, 65536, 65536, 65536, 1048576, 1048576, 1048576, 1048576, 4194304, 4194304, 4194304, 4194304, 16777216, 16777216, 16777216

Offset: 0

Tom Edgar, May 27 2014

nonn

This is the generalized factorial for A234957.

a(0) = 1 as it represents the empty product.

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. A054893, A060818, A060828, A234957.

S=[0]+[4^valuation(i,4) for i in [1..100]]
[prod(S[1:i+1]) for i in [0..99]]

a(n) = Product_{i=1..n} A234957(i).

a(n) = 4^(A054893(n)). - Vaclav Kotesovec, May 28 2014

A243757 a(n) = Product_{i=1..n} A060904(i).

Original entry on oeis.org

1, 1, 1, 1, 1, 5, 5, 5, 5, 5, 25, 25, 25, 25, 25, 125, 125, 125, 125, 125, 625, 625, 625, 625, 625, 15625, 15625, 15625, 15625, 15625, 78125, 78125, 78125, 78125, 78125, 390625, 390625, 390625, 390625, 390625, 1953125, 1953125, 1953125, 1953125, 1953125, 9765625

Offset: 0

Tom Edgar, Jun 10 2014

nonn

This is the generalized factorial for A060904.
a(0) = 1 as it represents the empty product.
a(n) is the largest power of 5 that divides n!, or the order of a 5-Sylow subgroup of the symmetric group of degree n. - David Radcliffe, Sep 03 2021

Reinhard Zumkeller, Table of n, a(n) for n = 0..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. A027868, A060818, A060828, A060904, A242954.

a243757 n = a243757_list !! n
a243757_list = scanl (*) 1 a060904_list
-- Reinhard Zumkeller, Feb 04 2015

Table[Product[5^IntegerExponent[k, 5], {k, 1, n}], {n, 0, 20}] (* G. C. Greubel, Dec 24 2016 *)

a(n) = prod(k=1,n, 5^valuation(k,5)); \\ G. C. Greubel, Dec 24 2016

S=[0]+[5^valuation(i, 5) for i in [1..100]]
[prod(S[1:i+1]) for i in [0..99]]

a(n) = Product_{i=1..n} A060904(i).

a(n) = 5^(A027868(n)).

A243758 a(n) = Product_{i=1..n} A234959(i).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 6, 6, 6, 6, 6, 6, 36, 36, 36, 36, 36, 36, 216, 216, 216, 216, 216, 216, 1296, 1296, 1296, 1296, 1296, 1296, 7776, 7776, 7776, 7776, 7776, 7776, 279936, 279936, 279936, 279936, 279936, 279936, 1679616, 1679616, 1679616, 1679616, 1679616, 1679616, 10077696

Offset: 0

Tom Edgar, Jun 10 2014

This is the generalized factorial for A234959.

a(0) = 1 as it represents the empty product.

Reinhard Zumkeller, Table of n, a(n) for n = 0..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. A060828, A060818, A234959, A242954, A243757.

a243758 n = a243758_list !! n
a243758_list = scanl (*) 1 a234959_list
-- Reinhard Zumkeller, Feb 09 2015

Table[Product[6^IntegerExponent[k, 6], {k, 1, n}], {n, 0, 20}] (* G. C. Greubel, Dec 24 2016 *)

valp(n,p)=my(s); while(n\=p, s+=n); s
a(n)=6^valp(n,6) \\ Charles R Greathouse IV, Oct 03 2016

S=[0]+[6^valuation(i,6) for i in [1..100]]
[prod(S[1:i+1]) for i in [0..99]]

a(n) = Product_{i=1..n} A234959(i).

a(n) = 6^(A054895(n)).

A243759 Triangle T(m,k): exponent of the highest power of 3 dividing the binomial coefficient binomial(m,k).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 1, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 2, 1, 2, 2, 1, 2, 2, 0, 0, 0, 2, 1, 1, 2, 1, 1, 2, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 1, 1, 0, 2, 2, 1, 2, 2

Offset: 0

Tom Edgar, Jun 10 2014

T(m,k) is the number of 'carries' that occur when adding k and n-k in base 3 using the traditional addition algorithm.

			The triangle begins:
0,
0, 0,
0, 0, 0,
0, 1, 1, 0;
0, 0, 1, 0, 0;
0, 0, 0, 0, 0, 0;
0, 1, 1, 0, 1, 1, 0;
0, 0, 1, 0, 0, 1, 0, 0;
0, 0, 0, 0, 0, 0, 0, 0, 0;
0, 2, 2, 1, 2, 2, 1, 2, 2, 0;

Robert Israel, Table of n, a(n) for n = 0..10010
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.

Row sums: A249343.

Cf. A007318, A007949, A083093, A065040, A242849, A060828, A038500.

A243759:= (m,k) -> padic[ordp](binomial(m,k),3);
for m from 0 to 50 do
  seq(A243759(m,k),k=0..m)
od;   # Robert Israel, Jun 15 2014

T[m_, k_] := IntegerExponent[Binomial[m, k], 3];
Table[T[m, k], {m, 0, 12}, {k, 0, m}] // Flatten (* Jean-François Alcover, Jun 05 2022 *)

m=50
T=[0]+[3^valuation(i, 3) 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]]
[log(Integer(x),base=3) for sublist in Table for x in sublist]

(define (A243759 n) (A007949 (A007318 n))) ;; Antti Karttunen, Oct 28 2014

T(m,k) = log_3(A242849(m,k)).
From Antti Karttunen, Oct 28 2014: (Start)
a(n) = A007949(A007318(n)).
a(n) * A083093(n) = 0 and a(n) + A083093(n) > 0 for all n.
(End)

Name clarified by Antti Karttunen, Oct 28 2014

A118381 Largest 3-smooth number dividing n!.

Original entry on oeis.org

1, 2, 6, 24, 24, 144, 144, 1152, 10368, 20736, 20736, 248832, 248832, 497664, 1492992, 23887872, 23887872, 429981696, 429981696, 1719926784, 5159780352, 10319560704, 10319560704, 247669456896, 247669456896, 495338913792, 13374150672384, 53496602689536, 53496602689536

Offset: 1

Reinhard Zumkeller, May 16 2006

nonn

Amiram Eldar, Table of n, a(n) for n = 1..1861 (terms below 10^1000)
Index entries for sequences related to factorial numbers.

Cf. A003586, A000142, A011371, A054861, A060818, A060828.

s[n_, b_] := Sum[IntegerExponent[k, b], {k, 1, n}]; a[n_] := 2^s[n, 2] * 3^s[n, 3]; Array[a, 30] (* Amiram Eldar, Jan 29 2020 *)

a(n) = A060818(n) * A060828(n).

More terms from Amiram Eldar, Jan 29 2020

A212307 Numerator of n!/3^n.

Original entry on oeis.org

1, 1, 2, 2, 8, 40, 80, 560, 4480, 4480, 44800, 492800, 1971200, 25625600, 358758400, 1793792000, 28700672000, 487911424000, 975822848000, 18540634112000, 370812682240000, 2595688775680000, 57105153064960000, 1313418520494080000, 10507348163952640000

Offset: 0

Jean-François Alcover and Paul Curtz, Oct 30 2013

Also the 3rd column of A152656 (or of A216919).

Cf. A001316, A049606, A125824 (denominators), A152656, A216919.

Cf. A000142, A060828.

Table[Numerator[n!/3^n], {n, 0, 32}]
(* or *) CoefficientList[Series[Exp[3x], {x, 0, 32}], x] // Denominator

a(n) = numerator(n!/3^n); \\ Michel Marcus, Oct 30 2013

a(n) = Product_{i=1..n} A038502(i). - Tom Edgar, Mar 22 2014

a(n) = A000142(n)/A060828(n). - Ridouane Oudra, Sep 23 2024

cp's OEIS Frontend

A242849 Triangle read by rows: T(n,k) = A060828(n)/(A060828(k) * A060828(n-k)).

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A136690 Final nonzero digit of n! in base 3.

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A090630 Greatest divisor d of n! such that d=m^k with k>1.

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A232097 a(n) = least k such that 1+2+3+...+k (k-th triangular number) is a multiple of n!; a(n) = least k such that A232096(k) >= n.

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A242954 a(n) = Product_{i=1..n} A234957(i).

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A243757 a(n) = Product_{i=1..n} A060904(i).

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A243758 a(n) = Product_{i=1..n} A234959(i).

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