A132022 -id:A132022 - cp's OEIS frontend

A132034 Decimal expansion of Product_{k>0} (1-1/6^k).

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

Offset: 0

Hieronymus Fischer, Aug 14 2007

			0.805687728162164940923750...

G. C. Greubel, Table of n, a(n) for n = 0..1200
Richard J. McIntosh, Some Asymptotic Formulae for q-Hypergeometric Series, Journal of the London Mathematical Society, Vol. 51, No. 1 (1995), pp. 120-136; alternative link.

Cf. A000203, A027873, A048651, A067080, A098844, A100220, A132022, A132026, A132038.

digits = 103; NProduct[1-1/6^k, {k, 1, Infinity}, NProductFactors -> 100, WorkingPrecision -> digits+20] // N[#, digits+20]& // RealDigits[#, 10, digits]& // First (* Jean-François Alcover, Feb 18 2014 *)
N[QPochhammer[1/6,1/6]] (* G. C. Greubel, Nov 30 2015 *)

prodinf(x=1, 1-(1/6)^x) \\ Altug Alkan, Dec 01 2015

Equals exp( -Sum_{n>0} sigma_1(n)/(n*6^n) ).
Equals (1/6; 1/6){infinity}, where (a;q){infinity} is the q-Pochhammer symbol. - G. C. Greubel, Nov 30 2015
From Amiram Eldar, May 09 2023: (Start)
Equals sqrt(2*Pi/log(6)) * exp(log(6)/24 - Pi^2/(6*log(6))) * Product_{k>=1} (1 - exp(-4*k*Pi^2/log(6))) (McIntosh, 1995).
Equals Sum_{n>=0} (-1)^n/A027873(n). (End)

A167747 a(n) = phi(6^n).

Original entry on oeis.org

1, 2, 12, 72, 432, 2592, 15552, 93312, 559872, 3359232, 20155392, 120932352, 725594112, 4353564672, 26121388032, 156728328192, 940369969152, 5642219814912, 33853318889472, 203119913336832, 1218719480020992, 7312316880125952, 43873901280755712, 263243407684534272

Offset: 0

Vladimir Joseph Stephan Orlovsky, Nov 10 2009

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
D. Bevan, D. Levin, P. Nugent, J. Pantone, L. Pudwell, M. Riehl and M. L. Tlachac, Pattern avoidance in forests of binary shrubs, arXiv preprint arXiv:1510:08036 [math.CO], 2015-2016.
Index entries for linear recurrences with constant coefficients, signature (6).

Cf. A000010, A000400, A132022.

Mathematica
```
Table[EulerPhi[6^n],{n,0,40}]
```

a(n) = eulerphi(6^n); \\ Michel Marcus, Jan 02 2021

a(n+1) = 2*6^n. - Charles R Greathouse IV, Nov 12 2009
G.f.: (1-4x)/(1-6x). - Philippe Deléham, Oct 10 2011
a(n) = ((8*n-4)*a(n-1)-12*(n-2)*a(n-2))/n, a(0)=1, a(1)=2. - Sergei N. Gladkovskii, Jul 19 2012
Sum_{n>=0} 1/a(n) = 8/5. - Amiram Eldar, Jan 02 2021
a(n) = A000010(A000400(n)). - Michel Marcus, Jan 02 2021
From Amiram Eldar, May 08 2023: (Start)
Sum_{n>=0} (-1)^n/a(n) = 4/7.
Product_{n>=1} (1 - 1/a(n)) = A132022. (End)

A132030 a(n) = Product_{k=0..floor(log_6(n))} floor(n/6^k), n>=1.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 24, 26, 28, 30, 32, 34, 54, 57, 60, 63, 66, 69, 96, 100, 104, 108, 112, 116, 150, 155, 160, 165, 170, 175, 216, 222, 228, 234, 240, 246, 294, 301, 308, 315, 322, 329, 384, 392, 400, 408, 416, 424, 486, 495, 504, 513, 522, 531, 600

Offset: 1

Hieronymus Fischer, Aug 20 2007

nonn

If n is written in base 6 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(52) = floor(52/6^0)*floor(52/6^1)*floor(52/6^2) = 52*8*1 = 416;
a(58) = 522 since 58 = 134_6 and so a(58) = 134_6 * 13_6 * 1_6 = 58*9*1 = 522.

G. C. Greubel, Table of n, a(n) for n = 1..10000

Cf. A048651, A132022, A132034, 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), 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.

f:= proc(n) option remember; n*procname(floor(n/6)) end proc:
f(0):= 1:
seq(f(i),i=1..100); # Robert Israel, Dec 20 2015

Table[Product[Floor[n/6^k], {k, 0, Floor[Log[6, n]]}], {n, 1, 100}] (* G. C. Greubel, Dec 20 2015 *)

Recurrence: a(n)=n*a(floor(n/6)); a(n*6^m)=n^m*6^(m(m+1)/2)*a(n).

a(k*6^m) = k^(m+1)*6^(m(m+1)/2), for 0

Asymptotic behavior: a(n) = O(n^((1+log_6(n))/2)); this follows from the inequalities below.

a(n) <= b(n), where b(n) = n^(1+floor(log_6(n)))/6^((1+floor(log_6(n)))*floor(log_6(n))/2); equality holds for n=k*6^m, 0=0. b(n) can also be written n^(1+floor(log_6(n)))/6^A000217(floor(log_6(n))).

Also: a(n) <= 2^((1-log_6(2))/2)*n^((1+log_6(n))/2) = 1.236766885...*6^A000217(log_6(n)), equality holds for n=2*6^m and for n=3*6^m, m>=0 (consider 2^((1-log_6(2))/2)=3^((1-log_6(3))/2) since 6=2*3).

a(n) > c*b(n), where c = 0.45071262522603913... (see constant A132022).

Also: a(n) > c*(sqrt(2)/2^log_6(sqrt(2)))*n^((1+log_6(n))/2) = 0.557426449...*6^A000217(log_6(n)).

lim inf a(n)/b(n) = 0.45071262522603913..., for n-->oo.

lim sup a(n)/b(n) = 1, for n-->oo.

lim inf a(n)/n^((1+log_6(n))/2) = 0.45071262522603913...*sqrt(2)/2^log_6(sqrt(2)), for n-->oo.

lim sup a(n)/n^((1+log_6(n))/2) = sqrt(3)/3^log_6(sqrt(3))=1.236766885..., for n-->oo.

lim inf a(n)/a(n+1) = 0.45071262522603913... for n-->oo (see constant A132022).

G.f. g(x) satisfies g(x) = (x+2x^2+3x^3+4x^4+5x^5)*(1 + g(x^6)) + 6*(x^6+x^7+x^8+x^9+x^10+x^11)*g'(x^6). - Robert Israel, Dec 20 2015

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A132034 Decimal expansion of Product_{k>0} (1-1/6^k).

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A167747 a(n) = phi(6^n).

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A132030 a(n) = Product_{k=0..floor(log_6(n))} floor(n/6^k), n>=1.

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