A132028 -id:A132028 - 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

A132027 a(n) = Product_{k=0..floor(log_3(n))} floor(n/3^k), n>=1.

Original entry on oeis.org

1, 2, 3, 4, 5, 12, 14, 16, 27, 30, 33, 48, 52, 56, 75, 80, 85, 216, 228, 240, 294, 308, 322, 384, 400, 416, 729, 756, 783, 900, 930, 960, 1089, 1122, 1155, 1728, 1776, 1824, 2028, 2080, 2132, 2352, 2408, 2464, 3375, 3450, 3525, 3840, 3920, 4000, 4335

Offset: 1

Hieronymus Fischer, Aug 13 2007, Aug 20 2007

nonn

If n is written in base 3 as n=d(m)d(m-1)d(m-2)...d(2)d(1)d(0) (where d(k) is the digit at position k) then a(n) is also the product d(m)d(m-1)d(m-2)...d(2)d(1)d(0)*d(m)d(m-1)d(m-2)...d(2)d(1)*d(m)d(m-1)d(m= -2)...d(2)*...*d(m)d(m-1)d(m-2)*d(m)d(m-1)*d(m).

			a(11)=floor(11/3^0)*floor(11/3^1)*floor(11/3^2)=11*3*1=33;
a(13)=52 since 13=111(base-3) and so a(13)=111*11*1(base-3)=13*4*1=52.

Ivan Neretin, Table of n, a(n) for n = 1..10000

Cf. A048651, A098844, A067080, A132019, A100220, A000217.
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=2 to p=12 see A098844(p=2), A132028(p=4)-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.

Table[(f = If[# < 3, #, #*f[Quotient[#, 3]]] &)[n], {n, 51}] (* Ivan Neretin, May 29 2016 *)

Recurrence: a(n)=n*a(floor(n/3)); a(n*3^m)=n^m*3^(m(m+1)/2)*a(n).
a(k*3^m)=k^(m+1)*3^(m(m+1)/2), for k=1 or 2.
a(n)<=b(n), where b(n)=n^(1+floor(log_3(n)))/3^(1/2*(1+floor(log_3(n)))*floor(log_3(n))); equality holds if n is a power of 3 or two times a power of 3.
Also: a(n)<=2^((1-log_3(2))/2)*n^((1+log_3(n))/2)=1.1364507...*3^A000217(log_3(n)), equality for n=2*3^m, m>=0.
a(n)>c*b(n), where c=0.3826631966790330232889550... (see constant A132019).
Also: a(n)>c*2^((1-log_3(2))/2)*n^((1+log_3(n))/2)=0.434877...*3^A000217(log_3(n)).
lim inf a(n)/b(n)=0.3826631966790330232889550..., for n-->oo.
lim sup a(n)/b(n)=1, for n-->oo.
lim inf a(n)/n^((1+log_3(n))/2)=0.3826631966790330232889550...*sqrt(2)/2^log_3(sqrt(2)), for n-->oo.
lim sup a(n)/n^((1+log_3(n))/2)=sqrt(2)/2^log_3(sqrt(2)), for n-->oo.
lim inf a(n)/a(n+1)=0.3826631966790330232889550... for n-->oo (see constant A132019).
a(n)=O(n^((1+log_3(n))/2)).

A132020 Decimal expansion of Product_{k>=0} (1 - 1/(2*4^k)).

Original entry on oeis.org

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

Offset: 0

Hieronymus Fischer, Aug 14 2007

This is the limiting probability that a large random symmetric binary matrix is nonsingular (cf. A086812, A048651). In other words, equals Lim_{n->oo} A086812(n)/A006125(n+1).- H. Tracy Hall, Sep 07 2024

			0.41942244179510759770995610770297425223395323439266674908044991663177...

John E. Cremona and R. W. K. Odoni, Some density results for negative Pell equations; an application of graph theory, Journal of the London Mathematical Society s2-39:1 (1989), pp. 16-28.

Cf. A004171, A048651, A098844, A067080, A132019, A132026, A132028, A100221, A000217, A081845.

evalf(1+sum((-1)^n*2^(n*(n-1)/2)/product(2^k-1, k=1..n), n=1..infinity), 120); # Robert FERREOL, Feb 23 2020

RealDigits[ Product[1 - 1/(2*4^i), {i, 0, 175}], 10, 111][[1]] (* Robert G. Wilson v, May 25 2011 *)
RealDigits[QPochhammer[1/2, 1/4], 10, 105][[1]] (* Jean-François Alcover, Nov 18 2015 *)

prodinf(k=0,1-1.>>(2*k+1)) \\ Charles R Greathouse IV, Nov 16 2012

Equals lim inf_{n->oo} Product_{k=0..floor(log_4(n))} floor(n/4^k)*4^k/n.
Equals lim inf_{n->oo} A132028(n)/n^(1+floor(log_4(n)))*4^((1/2)*(1+floor(log_4(n)))*floor(log_4(n))).
Equals lim inf_{n->oo} A132028(n)/n^(1+floor(log_4(n)))*4^A000217(floor(log_4(n))).
Equals (1/2)*exp(-Sum_{n>0} (4^(-n)*(Sum_{k|n} 1/(k*2^k)))).
Equals lim inf_{n->oo} A132028(n)/A132028(n+1).
Equals Product_{k>0} (1-1/(2^k+1)). - Robert G. Wilson v, May 25 2011
From Robert FERREOL, Feb 23 2020: (Start)
Equals Product_{k>0} (1 + 1/2^k)^(-1) = 2/A081845.
Equals 1 + Sum_{n>=1} (-1)^n*2^(n*(n-1)/2)/((2-1)*(2^2-1)*...*(2^n-1)). (End)
From Peter Bala, Jan 15 2021: (Start)
Constant C = Sum_{n >= 0} 2^n/Product_{k = 1..n} (1 - 4^k).
Faster converging series:
2*C = (1/2)*Sum_{n >= 0} 2^(-n)/Product_{k = 1..n} (1 - 4^k);
(2^4)*C = 7*Sum_{n >= 0} 2^(-3*n)/Product_{k = 1..n} (1 - 4^k);
(2^9)*C = 7*31*Sum_{n >= 0} 2^(-5*n)/Product_{k = 1..n} (1 - 4^k), and so on.
Slower converging series:
C = -Sum_{n >= 0} 2^(3*n)/Product_{k = 1..n} (1 - 4^k);
7*C = Sum_{n >= 0} 2^(5*n)/Product_{k = 1..n} (1 - 4^k);
7*31*C = -Sum_{n >= 0} 2^(7*n)/Product_{k = 1..n} (1 - 4^k), and so on. (End)
Equals Product_{n>=0} (1 - 1/A004171(n)). - Amiram Eldar, May 09 2023

Name corrected by Charles R Greathouse IV, Nov 16 2012

A054893 a(n) = Sum_{j > 0} floor(n/4^j).

Original entry on oeis.org

0, 0, 0, 0, 1, 1, 1, 1, 2, 2, 2, 2, 3, 3, 3, 3, 5, 5, 5, 5, 6, 6, 6, 6, 7, 7, 7, 7, 8, 8, 8, 8, 10, 10, 10, 10, 11, 11, 11, 11, 12, 12, 12, 12, 13, 13, 13, 13, 15, 15, 15, 15, 16, 16, 16, 16, 17, 17, 17, 17, 18, 18, 18, 18, 21, 21, 21, 21, 22, 22, 22, 22, 23, 23, 23, 23, 24, 24, 24, 24

Offset: 0

Henry Bottomley, May 23 2000

Different from highest power of 4 dividing n! (see A090616).

			  a(10^0) = 0.
  a(10^1) = 2.
  a(10^2) = 32.
  a(10^3) = 330.
  a(10^4) = 3331.
  a(10^5) = 33330.
  a(10^6) = 333330.
  a(10^7) = 3333329.
  a(10^8) = 33333328.
  a(10^9) = 333333326.

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

Cf. A053737, A235127 (first differences).

Cf. A011371, A027868, A054861, A054899, A067080, A090616, A098844, A132028.

function A054893(n)
  if n eq 0 then return n;
  else return A054893(Floor(n/4)) + Floor(n/4);
  end if; return A054893;
end function;
[A054893(n): n in [0..103]]; // G. C. Greubel, Feb 09 2023

Table[t=0; p=4; While[s=Floor[n/p]; t=t+s; s>0, p *= 4]; t, {n,0,100}]
Table[Total[Floor/@(n/NestList[4#&,4,6])],{n,0,80}] (* Harvey P. Dale, Jun 12 2022 *)

a(n) = (n - sumdigits(n,4))/3; \\ Kevin Ryde, Jan 08 2024

def A054893(n):
    if (n==0): return 0
    else: return A054893(n//4) + (n//4)
[A054893(n) for n in range(104)] # G. C. Greubel, Feb 09 2023

a(n) = floor(n/4) + floor(n/16) + floor(n/64) + floor(n/256) + ...
a(n) = (n - A053737(n))/3.
From Hieronymus Fischer, Sep 15 2007: (Start)
a(n) = a(floor(n/4)) + floor(n/4).
a(4*n) = a(n) + n.
a(n*4^m) = a(n) + n*(4^m-1)/3.
a(k*4^m) = k*(4^m-1)/3, for 0 <= k < 4, m >= 0.
Asymptotic behavior:
a(n) = n/3 + O(log(n)),
a(n+1) - a(n) = O(log(n)); this follows from the inequalities below.
a(n) <= (n-1)/3; equality holds true for powers of 4.
a(n) >= (n-3)/3 - floor(log_4(n)); equality holds true for n = 4^m - 1, m>0. lim inf (n/3 - a(n)) = 1/3, for n-->oo.
lim sup (n/3 - log_4(n) - a(n)) = 0, for n-->oo.
lim sup (a(n+1) - a(n) - log_4(n)) = 0, for n-->oo.
G.f.: (1/(1-x))*Sum_{k > 0} x^(4^k)/(1-x^(4^k)). (End)
Partial sums of A235127. - R. J. Mathar, Jul 08 2021

Edited by Hieronymus Fischer, Sep 15 2007

Examples added by Hieronymus Fischer, Jun 06 2012

A233035 a(n) = n * floor(n/4).

Original entry on oeis.org

0, 0, 0, 4, 5, 6, 7, 16, 18, 20, 22, 36, 39, 42, 45, 64, 68, 72, 76, 100, 105, 110, 115, 144, 150, 156, 162, 196, 203, 210, 217, 256, 264, 272, 280, 324, 333, 342, 351, 400, 410, 420, 430, 484, 495, 506, 517, 576, 588, 600, 612, 676, 689, 702, 715, 784, 798, 812, 826, 900, 915, 930, 945, 1024, 1040

Offset: 1

Kival Ngaokrajang, Dec 03 2013

The maximum number of I patterns tetrominos that can be packed into an n X n array of squares with rotation is prohibited.
u(n) = n*(n mod 4), where u(n) is total number of squares left after packing I patterns into n X n squares.
a(n) = A132028(n) for 4 <= n <= 31.

Kival Ngaokrajang, Illustration of initial terms
Wikipedia, Tetromino
Index entries for linear recurrences with constant coefficients, signature (1,0,0,2,-2,0,0,-1,1).

Cf. A132028, A233036, A242669.

Table[n*Floor[n/4],{n,80}] (* or *) LinearRecurrence[{1,0,0,2,-2,0,0,-1,1},{0,0,0,4,5,6,7,16,18},80] (* Harvey P. Dale, Aug 22 2020 *)

a(n) = n * floor(n/4); \\ Joerg Arndt, Dec 08 2013

a(n) = (n^2 - n*(n mod 4))/4.

G.f.: (x^7 + x^6 + x^5 + x^4 + 4*x^3)/((1-x)*(1-x^4)^2). - Ralf Stephan, Dec 08 2013

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

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A132027 a(n) = Product_{k=0..floor(log_3(n))} floor(n/3^k), n>=1.

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A132020 Decimal expansion of Product_{k>=0} (1 - 1/(2*4^k)).

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A054893 a(n) = Sum_{j > 0} floor(n/4^j).

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A233035 a(n) = n * floor(n/4).

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