A132030 -id:A132030 - cp's OEIS frontend

A098844 a(1)=1, a(n) = n*a(floor(n/2)).

1, 2, 3, 8, 10, 18, 21, 64, 72, 100, 110, 216, 234, 294, 315, 1024, 1088, 1296, 1368, 2000, 2100, 2420, 2530, 5184, 5400, 6084, 6318, 8232, 8526, 9450, 9765, 32768, 33792, 36992, 38080, 46656, 47952, 51984, 53352, 80000, 82000, 88200, 90300

Offset: 1

Benoit Cloitre, Nov 03 2004

nonn

			a(10) = floor(10/2^0)*floor(10/2^1)*floor(10/2^2)*floor(10/2^3) = 10*5*2*1 = 100;
a(17) = 1088 since 17 = 10001(base 2) and so a(17) = 10001*1000*100*10*1(base 2) = 17*8*4*2*1 = 1088.

Hieronymus Fischer, Table of n, a(n) for n = 1..1000

Cf. A048651, A067080, A132027, A132028, A132029, A132030, A132019, A132026, A132038.
For formulas regarding a general parameter p (i.e., terms floor(n/p^k)) see A132264.
For the product of terms floor(n/p^k) for p=3 to p=12 see A132027(p=3)-A132033(p=9), A067080(p=10), A132263(p=11), A132264(p=12).
For the products of terms 1+floor(n/p^k) see A132269-A132272, A132327, A132328.

lst={};Do[p=n;s=1;While[p>1,p=IntegerPart[p/2];s*=p;];AppendTo[lst,s],{n,1,6!,2}];lst (* Vladimir Joseph Stephan Orlovsky, Jul 28 2009 *)

PARI
```
a(n)=if(n<2,1,n*a(floor(n/2)))
```

from math import prod
def A098844(n): return n*prod(n//2**k for k in range(1,n.bit_length()-1)) # Chai Wah Wu, Jun 07 2022

a(2^n) = 2^(n*(n+1)/2) = A006125(n+1).
From Hieronymus Fischer, Aug 13 2007: (Start)
a(n) = Product_{k=0..floor(log_2(n))} floor(n/2^k), n>=1.
Recurrence:
a(n*2^m) = n^m*2^(m(m+1)/2)*a(n).
a(n) <= n^((1+log_2(n))/2) = 2^A000217(log_2(n)); equality iff n is a power of 2.
a(n) >= c(n)*(n+1)^((1 + log_2(n+1))/2) for n != 2,
where c(n) = Product_{k=1..floor(log_2(n))} (1 - 1/2^k); equality holds iff n+1 is a power of 2.
a(n) > c*(n+1)^((1 + log_2(n+1))/2)
where c = 0.288788095086602421... (see constant A048651).
lim inf a(n)/n^((1+log_2(n))/2)=0.288788095086602421... for n-->oo.
lim sup a(n)/n^((1+log_2(n))/2) = 1 for n-->oo.
lim inf a(n)/a(n+1) = 0.288788095086602421... for n-->oo (see constant A048651).
a(n) = O(n^((1+log_2(n))/2)). (End)

Formula section edited by Hieronymus Fischer, Jun 13 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

A132022 Decimal expansion of Product_{k>=0} (1 - 1/(2*6^k)).

Original entry on oeis.org

4, 5, 0, 7, 1, 2, 6, 2, 5, 2, 2, 6, 0, 3, 9, 1, 3, 0, 8, 3, 0, 0, 0, 0, 7, 8, 9, 5, 8, 3, 5, 2, 7, 1, 5, 5, 6, 0, 4, 4, 6, 7, 8, 5, 0, 0, 5, 4, 0, 0, 8, 5, 4, 7, 4, 3, 9, 0, 4, 5, 8, 3, 4, 8, 9, 2, 4, 4, 0, 9, 6, 0, 7, 5, 4, 0, 6, 2, 9, 4, 0, 7, 8, 2, 4, 3, 5, 3, 4, 5, 3, 1, 8, 6, 0, 8, 9, 6, 2, 6, 9, 2, 7

Offset: 0

Hieronymus Fischer, Aug 14 2007

			0.45071262522603913...

G. C. Greubel, Table of n, a(n) for n = 0..2000

Cf. A000217, A048651, A098844, A067080, A132019, A132026, A132030, A132034, A167747.

digits = 103; NProduct[1-1/(2*6^k), {k, 0, Infinity}, NProductFactors -> 200, WorkingPrecision -> digits+5] // N[#, digits+5]& // RealDigits[#, 10, digits]& // First (* Jean-François Alcover, Feb 18 2014 *)
(1/2)*N[QPochhammer[1/12, 1/6], 200] (* G. C. Greubel, Dec 20 2015 *)

prodinf(x=0, 1-1/(2*6^x)) \\ Altug Alkan, Dec 20 2015

Equals lim inf_{n->oo} Product_{k=0..floor(log_6(n))} floor(n/6^k)*6^k/n.
Equals lim inf_{n->oo} A132030(n)/n^(1+floor(log_6(n)))*6^(1/2*(1+floor(log_6(n)))*floor(log_6(n))).
Equals lim inf_{n->oo} A132030(n)/n^(1+floor(log_6(n)))*6^A000217(floor(log_6(n))).
Equals (1/2)*exp(-Sum_{n>0} 6^(-n)*Sum{k|n} 1/(k*2^k)).
Equals lim inf_{n->oo} A132030(n)/A132030(n+1).
Equals (1/2)*(1/12; 1/6){infinity}, where (a;q){infinity} is the q-Pochhammer symbol. - G. C. Greubel, Dec 20 2015
Equals Product_{n>=1} (1 - 1/A167747(n)). - Amiram Eldar, May 09 2023

cp's OEIS Frontend

A098844 a(1)=1, a(n) = n*a(floor(n/2)).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Python

Formula

Extensions

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Haskell

Magma

Mathematica

SageMath

Formula

Extensions

A132022 Decimal expansion of Product_{k>=0} (1 - 1/(2*6^k)).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula