A090618 -id:A090618 - cp's OEIS frontend

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

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).

A054898 a(n) = Sum_{k>0} floor(n/9^k).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 7, 7, 7, 7, 8, 8, 8, 8, 8, 8, 8, 8, 8, 10, 10, 10, 10, 10, 10, 10, 10, 10, 11, 11, 11, 11, 11, 11, 11

Offset: 0

Henry Bottomley, May 23 2000

nonn

Different from the highest power of 9 dividing n!, A090618.

			a(100)=12.
a(10^3)=124.
a(10^4)=1248.
a(10^5)=12498.
a(10^6)=124996.
a(10^7)=1249997.
a(10^8)=12499996.
a(10^9)=124999997.

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

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

Cf. A054899, A067080, A098844, A132033.

Table[t = 0; p = 9; While[s = Floor[n/p]; t = t + s; s > 0, p *= 9]; t, {n, 0, 100} ]
Table[Sum[Floor[n/9^k],{k,n}],{n,0,100}] (* Harvey P. Dale, Jul 10 2024 *)

a(n) = floor(n/9) + floor(n/81) + floor(n/729) + floor(n/6561) + ....
a(n) = (n-A053830(n))/8.
From Hieronymus Fischer, Aug 14 2007: (Start)
Recurrence:
a(n) = floor(n/9) + a(floor(n/9));
a(9*n) = n + a(n);
a(n*9^m) = n*(9^m-1)/8 + a(n).
a(k*9^m) = k*(9^m-1)/8, for 0<=k<9, m>=0.
Asymptotic behavior:
a(n) = n/8 + O(log(n)),
a(n+1) - a(n) = O(log(n)); this follows from the inequalities below.
a(n) <= (n-1)/8; equality holds for powers of 9.
a(n) >= (n-8)/8 - floor(log_9(n)); equality holds for n=9^m-1, m>0.
lim inf (n/8 - a(n)) =1/8, for n-->oo.
lim sup (n/8 - log_9(n) - a(n)) = 0, for n-->oo.
lim sup (a(n+1) - a(n) - log_9(n)) = 0, for n-->oo.
G.f.: g(x) = sum{k>0, x^(9^k)/(1-x^(9^k))}/(1-x). (End)

Examples added by Hieronymus Fischer, Jun 06 2012

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.

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.

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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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A054898 a(n) = Sum_{k>0} floor(n/9^k).

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A090619 Highest power of 12 dividing n!.

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A090621 Exponent of highest power of 16 dividing n!.

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