A090622 -id:A090622 - cp's OEIS frontend

A090624 If n = Product(pj^ej), i.e., written in its prime factorization, then a(n) = max_j{(pj-1)*ej}.

1, 2, 2, 4, 2, 6, 3, 4, 4, 10, 2, 12, 6, 4, 4, 16, 4, 18, 4, 6, 10, 22, 3, 8, 12, 6, 6, 28, 4, 30, 5, 10, 16, 6, 4, 36, 18, 12, 4, 40, 6, 42, 10, 4, 22, 46, 4, 12, 8, 16, 12, 52, 6, 10, 6, 18, 28, 58, 4, 60, 30, 6, 6, 12, 10, 66, 16, 22, 6, 70, 4, 72, 36, 8, 18, 10, 12, 78, 4, 8, 40, 82, 6

Offset: 2

Henry Bottomley, Dec 06 2003

nonn

The highest power of k dividing n! (A090622) is close to, but below, n/a(k).

			72 = 2^3*3^2 so a(72) = max((2-1)*3, (3-1)*2) = max(3,4) = 4.

David W. Wilson, Table of n, a(n) for n = 2..10000
A. M. Oller-Marcen and J. Maria Grau, On the Base-b Expansion of the Number of Trailing Zeros of b^k!, J. Int. Seq. 14 (2011) 11.6.8, propos. 3.

seq(max(map(t ->(t[1]-1)*t[2], ifactors(n)[2])),n=2..100); # Robert Israel, Sep 06 2016

a[n_] := Module[{aux = FactorInteger[n]},Max[Table[aux[[i, 2]]*(aux[[i, 1]] - 1), {i, 1, Length[aux]}]]] (* José María Grau Ribas, Feb 15 2010 *)

a(n)=my(f=factor(n)); vecmax(vector(#f~,i,(f[i,1]-1)*f[i,2])) \\ Charles R Greathouse IV, Sep 07 2016

from sympy import factorint
def A090624(n): return max((p-1)*e for p, e in factorint(n).items()) # Chai Wah Wu, Apr 28 2023

a(p) = p-1; a(p^m) = (p-1)*m.
a(b*c) = max(a(b), a(c)) for b and c coprime.
a(n) = lim_{k->inf} k/A090622(k, n) = lim_{k->inf} (k/highest power of k dividing n!). - David W. Wilson, Sep 05 2016

A090618 Highest power of 9 dividing n!.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 1, 1, 2, 2, 2, 2, 2, 2, 3, 3, 3, 4, 4, 4, 4, 4, 4, 5, 5, 5, 6, 6, 6, 7, 7, 7, 7, 7, 7, 8, 8, 8, 9, 9, 9, 9, 9, 9, 10, 10, 10, 11, 11, 11, 11, 11, 11, 13, 13, 13, 13, 13, 13, 14, 14, 14, 15, 15, 15, 15, 15, 15, 16, 16, 16, 17, 17, 17, 17, 17, 17, 18, 18, 18, 20, 20, 20, 20

Offset: 0

Henry Bottomley, Dec 06 2003

nonn

			a(9)=2 since 9!=362880=9^2*4480.

Cf. A011371, A054861, A090616, A027868, A054896, A090617.

IntegerExponent[Range[0,90]!,9] (* Harvey P. Dale, Jun 07 2016 *)

a(n) =A090622(n, 9) =[A054861(n)/2] =[([n/3]+[n/9]+[n/27]+[n/81]+...)/2].

A090619 Highest power of 12 dividing n!.

Original entry on oeis.org

0, 0, 0, 0, 1, 1, 2, 2, 2, 3, 4, 4, 5, 5, 5, 5, 6, 6, 8, 8, 8, 9, 9, 9, 10, 10, 10, 11, 12, 12, 13, 13, 14, 15, 15, 15, 17, 17, 17, 17, 18, 18, 19, 19, 19, 20, 21, 21, 22, 22, 22, 23, 23, 23, 25, 25, 26, 26, 27, 27, 28, 28, 28, 28, 30, 30, 31, 31, 31, 32, 32, 32, 34, 34, 34, 35, 35

Offset: 0

Henry Bottomley, Dec 06 2003

nonn

Most sequences of the form "highest power of k dividing n!" essentially depend on one of the primes or prime powers dividing k. But in this case, the sequences with k=3 (A054861) and k=4 (A090616) are both close to n/2 and vary in which one is lower for different values of n.

a(2^n) = A090616(2^n) and a(3^n-1) = A090616(3^n-1) while a(2^n-1) = A054861(2^n-1) and a(3^n) = A054861(3^n). - Robert Israel, Mar 25 2018

			a(6)=2 since 6!=720=12^2*5.

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

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

f2:= n -> n - convert(convert(n,base,2),`+`):
f3:= n -> (n - convert(convert(n,base,3),`+`))/2:
f:= n -> min(f3(n), floor(f2(n)/2)):
f(0):= 0:
map(f, [$0..100]); # Robert Israel, Mar 23 2018

Table[IntegerExponent[n!, 12], {n, 0, 100}] (* Jean-François Alcover, Mar 26 2018 *)

a(n) = valuation(n!, 12); \\ Michel Marcus, Mar 24 2018

a(n) =A090622(n, 12) =min(A054861(n), A090616(n)). Close to n/2, indeed for n>3: n/2-log3(n+1) <= a(n) < n/2.

A058067 Number of polynomial functions from Z to Z/nZ.

Original entry on oeis.org

1, 1, 4, 27, 64, 3125, 108, 823543, 1024, 19683, 12500, 285311670611, 1728, 302875106592253, 3294172, 84375, 65536, 827240261886336764177, 78732, 1978419655660313589123979, 200000, 22235661, 1141246682444, 20880467999847912034355032910567

Offset: 0

N. J. A. Sloane, Nov 24 2000

The first formula for a(n) is due to Kempner (1921). - Jonathan Sondow, Nov 05 2017

Amiram Eldar, Table of n, a(n) for n = 0..388
Manjul Bhargava, The factorial function and generalizations, Amer. Math. Monthly, Vol. 107, No. 9 (Nov. 2000), pp. 783-799; alternative link,
Aubrey J. Kempner, Polynomials and their residue systems, continued, Amer. Math. Soc. Trans., Vol. 22 (1921), pp. 240-288.
David Singmaster, On polynomial functions (mod m), Journal of Number Theory, Volume 6, Issue 5, October 1974, pp. 345-352.

Cf. A051674, A090622, A240098.

A058067 := n->mul(n/gcd(n,k!),k=0..n-1);

a[0] = 1; a[n_] := Product[n/GCD[n, k!], {k, 0, n - 1}]; Array[a, 24, 0] (* Amiram Eldar, Sep 29 2020 *)

a(n) = prod(k=0, n-1, n/gcd(n, k!)); \\ Michel Marcus, Nov 06 2017

a(n) = Product_{k=0..n-1} n/gcd(n, k!).
Multiplicative with a(p^e) = p^t_p(e). - David W. Wilson, Aug 14 2005 [t_p(e) = Sum_{k>=0: e > A090622(k, p)} (e - A090622(k, p)) = p * Sum_{k = 1..e} max(0, k - A090622(e-k, p)). In particular, t_p(e) = p*e*(e+1)/2 for e <= p. - Andrey Zabolotskiy, Nov 09 2017 and Sep 29 2020]
a(prime(n)) = A051674(n). - R. J. Mathar, Apr 01 2014 [Edited by Andrey Zabolotskiy, Nov 08 2017]
a(n) = n^n / A240098(n). - Jonathan Sondow, Nov 10 2017

A217445 Numbers n such that n! has the same number of terminating zeros in bases 3 and 4.

Original entry on oeis.org

1, 2, 4, 5, 6, 7, 10, 11, 12, 13, 14, 18, 19, 21, 22, 23, 33, 36, 37, 38, 42, 43, 46, 47, 51, 56, 58, 59, 60, 61, 62, 75, 86, 88, 89, 92, 100, 101, 102, 103, 105, 112, 113, 114, 115, 120, 121, 122, 124, 125, 138, 139, 141, 147, 153, 159, 164, 166, 167, 168

Offset: 1

Tanya Khovanova, Oct 03 2012

The number of zeros of n! base 3 is approaching n/2 as n grows. Similarly, the number of zeros of n! base 4 is approaching n/2 as n grows. Consequently, this sequence is expected to have high density.
From Robert Israel, Jan 19 2017: (Start)
Numbers n such that A000120(n) + (n + A000120(n) mod 2) = A053735(n).
Since typically A000120(n) ~ log_2(n) while typically A053735(n) ~ log_3(n), the density of this sequence should go to 0, contrary to the previous comment. (End)
Comment from N. J. A. Sloane, Dec 06 2019: (Start)
Appears to be the same as the list of positive numbers n such that the last nonzero digit of n! in base 12 belongs to the set [1, 2, 5, 7, 10, 11].
The first footnote in Deshouillers et al. (2016) says: "if the last nonzero digit of n! in base 12 belongs to {1, 2, 5, 7, 10, 11} then |(digit-sum of n in base 3) - (digit-sum of n in base 2)| is <= 1; this seems to occur infinitely many times." Compare A096288. (End)

			6! is 222200 in base 3 and 23100 in base 4, both of them have 2 zeros at the end, so 6 is in the sequence.

Jean-Marc Deshouillers, Laurent Habsieger, Shanta Laishram, Bernard Landreau, Sums of the digits in bases 2 and 3, arXiv:1611.08180, 2016

Alois P. Heinz, Table of n, a(n) for n = 1..10000

Cf. A054861 (base 3), A090616 (base 4), A090622, A096288.

s2:= n -> convert(convert(n,base,2),`+`):
s3:= n -> convert(convert(n,base,3),`+`):
select(n -> s2(n) + (n+s2(n) mod 2) = s3(n), [$1..1000]); # Robert Israel, Jan 19 2017

sntzQ[n_]:=Module[{f=n!},Last[Split[IntegerDigits[f,3]]]==Last[ Split[ IntegerDigits[ f,4]]]]; Select[Range[200],sntzQ] (* Harvey P. Dale, Jul 11 2020 *)

is(n)=my(L=log(n+1));sum(k=1,L\log(3),n\3^k)==sum(k=1,L\log(2),n>>k)\2 \\ Charles R Greathouse IV, Oct 04 2012

More terms from Alois P. Heinz, Oct 03 2012

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.

A091136 Smallest number m such that number of times m divides k! is almost k/n for large k, i.e., smallest m with A090624(m)=n.

Original entry on oeis.org

1, 2, 3, 8, 5, 32, 7, 128, 25, 512, 11, 2048, 13, 8192, 2187, 32768, 17, 131072, 19, 524288, 121, 2097152, 23, 8388608, 169, 33554432, 1594323, 134217728, 29, 536870912, 31, 2147483648, 289, 8589934592, 129140163, 34359738368, 37

Offset: 0

Henry Bottomley, Dec 19 2003

nonn

			a(2)=3 noting that 100! is a multiple of 3^48 and 48 is almost 100/2.

Cf. A090622, A090624, A091137.

a(n) = min_p{p prime and n divisible by p-1} p^(n/(p-1)).
a(p-1) = p.
a(2n+1) = 2^(2n+1).
Smallest divisor of A091137(n) which is not a divisor of A091137(n-1).

cp's OEIS Frontend

A090624 If n = Product(pj^ej), i.e., written in its prime factorization, then a(n) = max_j{(pj-1)*ej}.

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A090618 Highest power of 9 dividing n!.

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A090619 Highest power of 12 dividing n!.

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A058067 Number of polynomial functions from Z to Z/nZ.

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A217445 Numbers n such that n! has the same number of terminating zeros in bases 3 and 4.

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A090621 Exponent of highest power of 16 dividing n!.

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A091136 Smallest number m such that number of times m divides k! is almost k/n for large k, i.e., smallest m with A090624(m)=n.

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