A054861 -id:A054861 - cp's OEIS frontend

A276624 The infinite trunk of ternary beanstalk with reversed subsections.

0, 2, 8, 4, 26, 20, 16, 12, 10, 80, 72, 68, 62, 56, 52, 46, 42, 38, 34, 30, 28, 242, 232, 224, 216, 212, 204, 198, 194, 188, 180, 176, 170, 164, 160, 152, 144, 140, 134, 126, 122, 116, 110, 106, 100, 96, 92, 88, 84, 82, 728, 716, 706, 698, 688, 680, 672, 664, 656, 648, 644, 634, 626, 618, 610, 602, 594, 590, 582, 576, 572

Offset: 0

Antti Karttunen, Sep 11 2016

See A276623.

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

Cf. A000244, A053735, A054861, A081604, A276623.

(definec (A276624 n) (cond ((zero? n) n) ((= n 1) 2) (else (let ((maybe_next (* 2 (A054861 (A276624 (- n 1)))))) (if (not (= 1 (A053735 (+ 1 maybe_next)))) maybe_next (+ -1 (A000244 (+ 1 (A081604 (+ 1 maybe_next))))))))))

a(0) = 0; a(1) = 2; for n > 1, if 2*A054861(a(n-1))+1 is not a power of 3, then a(n) = 2*A054861(a(n-1)), otherwise a(n) = A000244(1+A081604(1+2*A054861(a(n-1)))) - 1.

A344853 a(n) = n minus (sum of digits of n in base 3).

Original entry on oeis.org

0, 0, 0, 2, 2, 2, 4, 4, 4, 8, 8, 8, 10, 10, 10, 12, 12, 12, 16, 16, 16, 18, 18, 18, 20, 20, 20, 26, 26, 26, 28, 28, 28, 30, 30, 30, 34, 34, 34, 36, 36, 36, 38, 38, 38, 42, 42, 42, 44, 44, 44, 46, 46, 46, 52, 52, 52, 54, 54, 54, 56, 56, 56, 60, 60, 60, 62, 62, 62

Offset: 0

Thomas König, May 30 2021

All terms are even.

In all sequences of the form f(n) = n minus (sum of digits of n in base b), every term appears b times consecutively. Here b = 3, hence terms are entries of A346502 repeated 3 times. - Bernard Schott, Jul 21 2021

			a(20) = 20 - (2 + 0 + 2) = 16 because 20 is written as 202 in base 3.

Michael De Vlieger, Table of n, a(n) for n = 0..10000

Cf. A011371 (in base 2), A066568 (in base 10).

Cf. A004128, A053735, A054861, A346502.

a[n_] := n - Plus @@ IntegerDigits[n, 3]; Array[a, 70, 0] (* Amiram Eldar, May 30 2021 *)

a(n) = n - sumdigits(n, 3); \\ Michel Marcus, Jul 11 2021

a(n) = n - A053735(n).
a(n) = 2 * A054861(n).
a(n) = 2 * A004128(floor(n/3)).
a(3*n) = a(3*n+1) = a(3*n+2).

A376410 Number of integers whose arithmetic derivative (A003415) is equal to n!, the n-th factorial.

Original entry on oeis.org

0, 1, 4, 13, 40, 186, 952, 5533, 38719, 346207, 3130816, 34444968, 382437431, 4637235152

Offset: 2

Antti Karttunen, Nov 06 2024

For 1! = 1, there are an infinite number of integers k for which A003415(k) = 1 (namely, all the primes), therefore the starting offset is 2.
Like with A351029, also here most of the solutions seem to be squarefree semiprimes, counted by A062311.
Terms a(12)..a(15) were obtained by summing the corresponding terms of A062311 and A377986.

Index entries for sequences related to factorial numbers

Cf. A000142, A002620, A003415, A062311, A099302, A377986.
Cf. A011371, A054861, A027868, A054896.
Cf. also A351029, A369239.

\\ Slow program, for computing just a few terms:
A002620(n) = ((n^2)>>2);
A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A376410(n) = { my(g=n!); sum(k=1,A002620(g),A003415(k)==g); };

A376410(n) = AntiDeriv(n!);
AntiDeriv(n,startvlen=1,solsfilename="") = { my(v = vector(startvlen,i,2), ip = #v, r, c=0); while(1, r = A003415vrl(v,n); if(0==r, ip--, if(r > 1, c++; if(solsfilename!="", write(solsfilename, r*factorback(v)))); ip = #v); if(0==ip, v = vector(1+#v,i,2); ip = #v; if(A003415vec(v) > n, return(c)), v[ip] = nextprime(1+v[ip]); for(i=1+ip, #v, v[i]=v[i-1]))); };
A003415vec(tv) = { my(n=factorback(tv), s=0, m=1, spf); for(i=1,#tv,spf = tv[i]; n /= spf; s += m*n; m *= spf); (s); }; \\ Compute Arithmetic derivative from the vector of primes.
A003415vrl(pv,lim) = { my(n=factorback(pv), x=lim-n, s=0, m=1, spf, u=n); for(i=1,#pv,spf = pv[i]; u /= spf; s += m*u; m *= spf); if(((x/s)

a(n) = A099302(A000142(n)).
a(n) = Sum_{k=1..A002620(n!)} [A003415(k) = n!], where [ ] is the Iverson bracket.
a(n) = A062311(n) + A377986(n).

A090621 Exponent of highest power of 16 dividing n!.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 3, 3, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 6, 6, 6, 6, 7, 7, 8, 8, 8, 8, 8, 8, 9, 9, 9, 9, 10, 10, 10, 10, 11, 11, 11, 11, 12, 12, 12, 12, 13, 13, 13, 13, 14, 14, 14, 14, 15, 15, 16, 16, 16, 16, 16, 16, 17, 17, 17, 17, 18, 18, 18, 18, 19, 19, 19, 19

Offset: 0

Henry Bottomley, Dec 06 2003

nonn

			a(10)=2 since 10! = 3628800 = 16^2 * 14175.

Cf. A011371, A054861, A090616, A027868, A054896, A090617, A090618, A090619, A090620.

a(n) = valuation(n!, 16); \\ Michel Marcus, Jul 10 2022

a(n) = A090622(n, 16) = floor(A011371(n)/4) = floor(A090616(n)/2) = floor((floor(n/2) + floor(n/4) + floor(n/8) + floor(n/16) + ...)/4). Almost n/4.

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

A127427 a(n) = v_3( (6n)! ) - v_3( (3n)! ), where v_3(N) is the 3-adic valuation A007949(N).

Original entry on oeis.org

0, 1, 3, 4, 5, 8, 9, 10, 12, 13, 14, 16, 17, 18, 22, 23, 24, 26, 27, 28, 30, 31, 32, 35, 36, 37, 39, 40, 41, 43, 44, 45, 48, 49, 50, 52, 53, 54, 56, 57, 58, 63, 64, 65, 67, 68, 69, 71, 72, 73, 76, 77, 78, 80, 81, 82, 84, 85, 86, 89, 90, 91, 93, 94, 95, 97, 98, 99, 103, 104, 105, 107

Offset: 0

N. J. A. Sloane, Apr 02 2007

nonn

Sung-Hyuk Cha, On Integer Sequences Derived from Balanced k-ary Trees, Applied Mathematics in Electrical and Computer Engineering, 2012. - From _N. J. A. Sloane_, Jun 12 2012
Sung-Hyuk Cha, On Complete and Size Balanced k-ary Tree Integer Sequences, International Journal of Applied Mathematics and Informatics, Issue 2, Volume 6, 2012, pp. 67-75. - From _N. J. A. Sloane_, Dec 24 2012

Cf. A007949, A053735, A054861, A004128.

Essentially partial sums of A127427.

s[n_] := Plus @@ IntegerDigits[n, 3]; a[n_] := (3*n + s[3*n] - s[6*n])/2; Array[a, 100, 0] (* Amiram Eldar, Feb 21 2021 *)

a(n) = valuation((6*n)!, 3) - valuation((3*n)!, 3); \\ Michel Marcus, Jul 29 2017

a(n) - n = a( [(n+1)/3] ).

a(n) = (3*n + A053735(n) - A053735(6*n))/2. - Amiram Eldar, Feb 21 2021

A088244 Number of 3-smooth divisors of n!.

Original entry on oeis.org

1, 1, 2, 4, 8, 8, 15, 15, 24, 40, 45, 45, 66, 66, 72, 84, 112, 112, 153, 153, 171, 190, 200, 200, 253, 253, 264, 336, 364, 364, 405, 405, 480, 512, 528, 528, 630, 630, 648, 684, 741, 741, 800, 800, 840, 924, 946, 946, 1081, 1081, 1104, 1152, 1200, 1200

Offset: 0

Reinhard Zumkeller, Jan 23 2004

nonn

Amiram Eldar, Table of n, a(n) for n = 0..10000

Cf. A000005, A003586.
Cf. A000142, A072078.
Cf. A011371, A054861.

a[n_] := ((1 + IntegerExponent[n!, 2])*(1 + IntegerExponent[n!, 3])); Array[a, 100, 0] (* Amiram Eldar, Aug 30 2019 *)

a(n) = A072078(A000142(n)).
a(n) = a(n-1) iff gcd(n,6) = 1.
a(n) = (A011371(n)+1)*(A054861(n)+1).

A089643 3^a(n) divides C(3n,n); 3-adic valuation of A005809.

Original entry on oeis.org

0, 1, 1, 1, 2, 1, 1, 2, 2, 1, 2, 2, 2, 3, 1, 1, 2, 2, 1, 2, 2, 2, 3, 2, 2, 3, 3, 1, 2, 2, 2, 3, 2, 2, 3, 3, 2, 3, 3, 3, 4, 1, 1, 2, 2, 1, 2, 2, 2, 3, 2, 2, 3, 3, 1, 2, 2, 2, 3, 2, 2, 3, 3, 2, 3, 3, 3, 4, 2, 2, 3, 3, 2, 3, 3, 3, 4, 3, 3, 4, 4, 1, 2, 2, 2, 3, 2, 2, 3, 3, 2, 3, 3, 3, 4, 2, 2, 3, 3, 2, 3, 3, 3, 4, 3

Offset: 0

Benoit Cloitre, Jan 01 2004

Antti Karttunen, Table of n, a(n) for n = 0..19683

Cf. A005809, A007949, A053735, A054861.

Table[IntegerExponent[Binomial[3 n, n], 3], {n, 0, 104}] (* Michael De Vlieger, Jul 29 2017 *)

PARI
```
a(n)=valuation(binomial(3*n,n),3)
```

from sympy import binomial
def a007949(n): return 0 if n%3 else a007949(n//3) + 1
def a(n): return a007949(binomial(3*n, n))
print([a(n) for n in range(151)]) # Indranil Ghosh, Jul 29 2017

a(n) = A007949(A005809(n)). - Antti Karttunen, Jul 29 2017
a(n) = A054861(3*n) - A054861(2*n) - A054861(n). - David A. Corneth, Jul 29 2017
a(n) = A053735(2*n)/2. - Amiram Eldar, Feb 21 2021

A135710 Positive integers b such that more than one prime factor p of b attains the maximum of (p-1)*v_p(b) where v_p(b) is the valuation of b at p.

Original entry on oeis.org

12, 45, 80, 90, 144, 180, 189, 240, 360, 378, 448, 637, 720, 756, 945, 1274, 1344, 1512, 1625, 1728, 1890, 1911, 2025, 2240, 2548, 2673, 3024, 3185, 3250, 3780, 3822, 4032, 4050, 4875, 5096, 5346, 5733, 6048, 6125, 6370, 6400, 6500, 6517, 6720, 7007, 7560, 7644

Offset: 1

Noam D. Elkies, Nov 25 2007

Given b, the number of trailing zeros at the end of the base-b representation of x! is asymptotic to x/M where M is the maximum over p|b of (p-1)*v_p(b).
Usually only one prime p attains the maximum and then the number is v_p(x!)/v_p(b) for all but finitely many x.
But for b=12,45,80,90,..., at least two v_p(x!) must be computed. For example: if b=12 then for x=2006 there are 998 trailing zeros due to v_3 but for x=2007 there are 999 due to v_2.

			For b=90 we have (p-1)*v_p(b) = 1, 4, 4 for p = 2, 3, 5 respectively so the maximum of 4 is attained twice (p=3 and p=5).

Eryk LIPKA, Automaticity of the sequence of the last nonzero digits of n! in a fixed base, Journal de Théorie des Nombres de Bordeaux 31 (2019), 283-291. [See Theorem 3.7 on page 290, and consider the complementary sequence.] - Jean-Paul Allouche and Don Reble, Oct 22 2020.

Alois P. Heinz, Table of n, a(n) for n = 1..2000
J.-P. Allouche, J. Shallit, and R. Yassawi, How to prove that a sequence is not automatic, arXiv:2104.13072 [math.NT], 2021.

Cf. A027868, A011371, A054861.

a:= proc(n) option remember; local k; for k from 1+
     `if`(n=1, 1, a(n-1)) while (s-> nops(s)<2 or (l->
      l[-2] (p-1)*padic[ordp](k, p),
       s))))([numtheory[factorset](k)[]]) do od; k
    end:
seq(a(n), n=1..50);  # Alois P. Heinz, Oct 23 2020

F[n_] := Module[{f, p, v, vmax}, f = FactorInteger[n]; p = f[[All, 1]]; v = Table[ f[[i, 2]]*(p[[i]]-1), {i, 1, Length[p]}]; vmax = Max[v]; Sum[Boole[v[[i]] == vmax], {i, 1, Length[v]}]]; Reap[For[n = 1, n <= 6400, n++, If[F[n] > 1, Print[n]; Sow[n]]]][[2, 1]] (* Jean-François Alcover, Jan 09 2014, translated from PARI *)

{ F(n, f,p,v,vmax)= f=factor(n); p=f[,1]; v=vector(length(p),i,f[i,2]*(p[i]-1)); vmax=vecmax(v); sum(i=1,length(v),v[i]==vmax) } for(n=1,6400,if(F(n)>1,print(n)))

A175417 Exponent of 2 minus sum of all other exponents, in the prime power factorization of n!

Original entry on oeis.org

0, 0, 1, 0, 2, 1, 1, 0, 3, 1, 1, 0, 1, 0, 0, -2, 2, 1, 0, -1, 0, -2, -2, -3, -1, -3, -3, -6, -5, -6, -7, -8, -3, -5, -5, -7, -7, -8, -8, -10, -8, -9, -10, -11, -10, -13, -13, -14, -11, -13, -14, -16, -15, -16, -18, -20, -18, -20, -20, -21, -21, -22, -22, -25, -19, -21, -22

Offset: 0

Zak Seidov, May 08 2010

sign

a(n)=0 for n={0,1,3,7,11,13,14,18,20}.

			a(20) = 0 because 20! = 2432902008176640000 = ((2^18)*(3^8)*(5^4)*(7^2)*(11^1)*(13^1)*(17^1)*(19^1)) and 18-(8+4+2+1+1+1+1) = 0.

Robert Israel, Table of n, a(n) for n = 0..10000

Cf. A000142, A011371, A027868, A054861, A054896.

f:= proc(n) local t,p,k;
      p:= 2: t:= add(floor(n/2^k),k=1..ilog2(n)):
      do
        p:= nextprime(p);
        if n < p then return t fi;
        t:= t - add(floor(n/p^k),k=1..ilog[p](n))
      od
end proc:
map(f, [$0..100]); # Robert Israel, Nov 10 2024

Table[2*IntegerExponent[m!,2]-Total[Last/@FactorInteger[m! ]],{m,0,130}]

a(n)=2*A011371(n)-A022559(n).

cp's OEIS Frontend

A276624 The infinite trunk of ternary beanstalk with reversed subsections.

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A344853 a(n) = n minus (sum of digits of n in base 3).

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A376410 Number of integers whose arithmetic derivative (A003415) is equal to n!, the n-th factorial.

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A090621 Exponent of highest power of 16 dividing n!.

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A118381 Largest 3-smooth number dividing n!.

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A127427 a(n) = v_3( (6n)! ) - v_3( (3n)! ), where v_3(N) is the 3-adic valuation A007949(N).

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A088244 Number of 3-smooth divisors of n!.

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A089643 3^a(n) divides C(3n,n); 3-adic valuation of A005809.

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