A165424 -id:A165424 - cp's OEIS frontend

A078887 Decimal expansion of Sum {n>=0} 1/6^(2^n).

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

Offset: 0

Robert G. Wilson v, Dec 11 2002

			0.195216644757251284925...

Aubrey J. Kempner, On Transcendental Numbers, Transactions of the American Mathematical Society, volume 17, number 4, October 1916, pages 476-482.
Index entries for transcendental numbers

Cf. A165424.

Similar sums: A007404, A078885, A078585, A078886, A078888, A078889, A078890, A036987.

RealDigits[ N[ Sum[1/6^(2^n), {n, 0, Infinity}], 110]][[1]]

suminf(n=0, 1/6^(2^n)) \\ Michel Marcus, Nov 11 2020

Equals -Sum_{k>=1} mu(2*k)/(6^k - 1), where mu is the Möbius function (A008683). - Amiram Eldar, Jul 12 2020

A173501 a(1) = 2, a(2) = 3, a(n) = 6^(2^(n-3)) for n >= 3.

Original entry on oeis.org

2, 3, 6, 36, 1296, 1679616, 2821109907456, 7958661109946400884391936, 63340286662973277706162286946811886609896461828096

Offset: 1

Giovanni Teofilatto, Feb 20 2010

nonn

Except for first three terms, a(n) = (a(n-1))^2.

Essentially the same as A165424. - R. J. Mathar, Feb 21 2010

G. C. Greubel, Table of n, a(n) for n = 1..13

Cf. A000079, A000400, A165424.

Table[If[n<3, n+1, 6^(2^(n-3))], {n,10}] (* G. C. Greubel, Apr 25 2021 *)

a(n) = if(n<=2, n+1, 6^(2^(n-3))) \\ Felix Fröhlich, Apr 25 2021

[2,3]+[6^(2^(n-3)) for n in (3..10)] # G. C. Greubel, Apr 25 2021

a(n) = Product_{i=1..n-1} a(i) with a(1) = 2 and a(2) = 3.

a(n) = 6^(2^(n-3)) = A000400(A000079(n-3)) for n>2.

Definition corrected by Glenn Tesler, Aug 19 2017

A225158 Denominators of the sequence of fractions f(n) defined recursively by f(1) = 6/1; f(n+1) is chosen so that the sum and the product of the first n terms of the sequence are equal.

Original entry on oeis.org

1, 5, 31, 1141, 1502761, 2555339110801, 7279526598745139799221281, 58396508924557918552199410007906486608310469119041, 3723292553725227196293782783863296586090351965218332181732394788182320381276998127547535467381368961

Offset: 1

Martin Renner, Apr 30 2013

Numerators of the sequence of fractions f(n) is A165424(n+1), hence sum(A165424(i+1)/a(i),i=1..n) = product(A165424(i+1)/a(i),i=1..n) = A165424(n+2)/A225165(n) = A173501(n+2)/A225165(n).

			f(n) = 6, 6/5, 36/31, 1296/1141, ...
6 + 6/5 = 6 * 6/5 = 36/5; 6 + 6/5 + 36/31 = 6 * 6/5 * 36/31 = 1296/155; ...

Cf. A100441, A165424, A173501, A225165.

b:=n->6^(2^(n-2)); # n > 1
b(1):=6;
p:=proc(n) option remember; p(n-1)*a(n-1); end;
p(1):=1;
a:=proc(n) option remember; b(n)-p(n); end;
a(1):=1;
seq(a(i),i=1..9);

a(n) = 6^(2^(n-2)) - product(a(i),i=1..n-1), n > 1 and a(1) = 1.

a(n) = 6^(2^(n-2)) - p(n) with a(1) = 1 and p(n) = p(n-1)*a(n-1) with p(1) = 1.

A225165 Denominators of the sequence s(n) of the sum resp. product of fractions f(n) defined recursively by f(1) = 6/1; f(n+1) is chosen so that the sum and the product of the first n terms of the sequence are equal.

Original entry on oeis.org

1, 5, 155, 176855, 265770796655, 679134511201261085170655, 4943777738415359153962876938905400001585992709055

Offset: 1

Martin Renner, Apr 30 2013

nonn

Numerators of the sequence s(n) of the sum resp. product of fractions f(n) is A165424(n+2), hence sum(A165424(i+1)/A225158(i),i=1..n) = product(A165424(i+1)/A225158(i),i=1..n) = A165424(n+2)/a(n) = A173501(n+2)/a(n).

			f(n) = 6, 6/5, 36/31, 1296/1141, ...
6 + 6/5 = 6 * 6/5 = 36/5; 6 + 6/5 + 36/31 = 6 * 6/5 * 36/31 = 1296/155; ...
s(n) = 1/b(n) = 6, 36/5, 1296/155, ...

Cf. A076628, A165424, A173501, A225158.

b:=proc(n) option remember; b(n-1)-b(n-1)^2; end:
b(1):=1/6;
a:=n->6^(2^(n-1))*b(n);
seq(a(i),i=1..8);

a(n) = 6^(2^(n-1))*b(n) where b(n)=b(n-1)-b(n-1)^2 with b(1)=1/6.

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A078887 Decimal expansion of Sum {n>=0} 1/6^(2^n).

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A173501 a(1) = 2, a(2) = 3, a(n) = 6^(2^(n-3)) for n >= 3.

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A225158 Denominators of the sequence of fractions f(n) defined recursively by f(1) = 6/1; f(n+1) is chosen so that the sum and the product of the first n terms of the sequence are equal.

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A225165 Denominators of the sequence s(n) of the sum resp. product of fractions f(n) defined recursively by f(1) = 6/1; f(n+1) is chosen so that the sum and the product of the first n terms of the sequence are equal.

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