A006276 -id:A006276 - cp's OEIS frontend

A091832 Pierce expansion of 1/e^2.

7, 18, 19, 136, 349, 357, 1354, 6996, 7135, 9531, 11558, 15996, 17432, 52118, 151048, 427802, 821834, 877819, 972918, 1046690, 1540789, 3653077, 8200738, 9628573, 164153335, 5607624822, 86457467082, 141885251873, 151882622551

Offset: 1

Benoit Cloitre, Mar 09 2004

nonn

If u(0) = exp(1/m) with m an integer >= 1 and u(n+1) = u(n)/frac(u(n)) then floor(u(n)) = m*n.

G. C. Greubel, Table of n, a(n) for n = 1..501 [a(1)=7 inserted by Georg Fischer, Nov 20 2020]
P. Erdős and Jeffrey Shallit, New bounds on the length of finite Pierce and Engel series, Sem. Théor. Nombres Bordeaux (2) 3 (1991), no. 1, 43-53.
Vlado Keselj, Length of Finite Pierce Series: Theoretical Analysis and Numerical Computations .
Jeffrey Shallit, Some predictable Pierce expansions, Fib. Quart., 22 (1984), 332-335.
Pelegrí Viader, Lluís Bibiloni, and Jaume Paradís, On a problem of Alfred Renyi, Economics Working Paper No. 340.

Cf. A006275, A006276, A006283.

Cf. A006784 (Pierce expansion definition), A059194.

PierceExp[A_, n_] := Join[Array[1 &, Floor[A]], First@Transpose@ NestList[{Floor[1/Expand[1 - #[[1]] #[[2]]]], Expand[1 - #[[1]] #[[2]]]} &, {Floor[1/(A - Floor[A])], A - Floor[A]}, n - 1]]; PierceExp[N[1/E^2, 7!], 15] (* G. C. Greubel, Nov 14 2016 *)

default(realprecision, 100000); r=exp(2); for(n=1, 100, s=(r/(r-floor(r))); print1(floor(r), ", "); r=s) \\ Benoit Cloitre [amended by Georg Fischer, Nov 20 2020]

Let u(0) = exp(2) and u(n+1) = u(n)/frac(u(n)) where frac(x) is the fractional part of x, then a(n) = floor(u(n)).
1/e^2 = 1/a(1) - 1/(a(1)*a(2)) + 1/(a(1)*a(2)*a(3)) - 1/(a(1)*a(2)*a(3)*a(4)) ...
Limit_{n->oo} a(n)^(1/n) = e.

a(1)=7 inserted by Georg Fischer, Nov 20 2020

A091833 Pierce expansion of 1/zeta(2).

Original entry on oeis.org

1, 2, 4, 7, 22, 29, 51, 173, 210, 262, 417, 746, 12341, 207220, 498538, 1286415, 2351289, 3702952, 7664494, 54693034, 75971438, 269954954, 6674693008, 13449203581, 59799655308, 98912303039, 948887634688, 3557757020909, 5898230078743

Offset: 1

Benoit Cloitre, Mar 09 2004

nonn

If u(0) = exp(1/m), m integer >= 1, and u(n+1) = u(n)/frac(u(n)) then floor(u(n)) = m*n.

G. C. Greubel, Table of n, a(n) for n = 1..1001 [a(1)=1 inserted by Georg Fischer, Nov 20 2020]
P. Erdős and Jeffrey Shallit, New bounds on the length of finite Pierce and Engel series, Sem. Théor. Nombres Bordeaux (2) 3 (1991), no. 1, 43-53.
Vlado Keselj, Length of Finite Pierce Series: Theoretical Analysis and Numerical Computations .
Jeffrey Shallit, Some predictable Pierce expansions, Fib. Quart., 22 (1984), 332-335.
Pelegrí Viader, Lluís Bibiloni, Jaume Paradís, On a problem of Alfred Renyi, Economics Working Paper No. 340.

Cf. A006275, A006276, A006283.

Cf. A006784 (Pierce expansion definition), A059186.

PierceExp[A_, n_] := Join[Array[1 &, Floor[A]], First@Transpose@ NestList[{Floor[1/Expand[1 - #[[1]] #[[2]]]], Expand[1 - #[[1]] #[[2]]]} &, {Floor[1/(A - Floor[A])], A - Floor[A]}, n - 1]]; PierceExp[N[1/Zeta[2], 7!], 25] (* G. C. Greubel, Nov 14 2016 *)

default(realprecision, 100000); r=zeta(2); for(n=1, 100, s=(r/(r-floor(r))); print1(floor(r), ", "); r=s) \\ Benoit Cloitre [amended by Georg Fischer, Nov 20 2020]

Let u(0) = Pi^2/6 and u(n+1) = u(n)/frac(u(n)) where frac(x) is the fractional part of x; then a(n) = floor(u(n)).
1/zeta(2) = 1/a(1) - 1/(a(1)*a(2)) + 1/(a(1)*a(2)*a(3)) - 1/(a(1)*a(2)*a(3)*a(4)) ...
Limit_{n->oo} a(n)^(1/n) = e.

a(1)=1 inserted by Georg Fischer, Nov 20 2020

A091846 Pierce expansion of log(2).

Original entry on oeis.org

1, 3, 12, 21, 51, 57, 73, 85, 96, 1388, 4117, 5268, 9842, 11850, 16192, 19667, 29713, 76283, 460550, 1333597, 1462506, 9400189, 13097390, 30254851, 190193800, 201892756, 431766247, 942050077, 6204785761, 16684400052, 23762490104

Offset: 1

Benoit Cloitre, Mar 09 2004

nonn

If u(0)=exp(1/m) m integer>=1 and u(n+1)=u(n)/frac(u(n)) then floor(u(n))=m*n.

G. C. Greubel, Table of n, a(n) for n = 1..500
P. Erdős and Jeffrey Shallit, New bounds on the length of finite Pierce and Engel series, Sem. Théor. Nombres Bordeaux (2) 3 (1991), no. 1, 43-53.
Vlado Keselj, Length of Finite Pierce Series: Theoretical Analysis and Numerical Computations .
Jeffrey Shallit, Some predictable Pierce expansions, Fib. Quart., 22 (1984), 332-335.
Pelegrí Viader, Lluís Bibiloni, Jaume Paradís, On a Problem of Alfred Renyi, Economics Working Paper No. 340.
Eric Weisstein's World of Mathematics, Pierce Expansion

Cf. A006275, A006276, A006283, A006284.

Cf. A006784 (Pierce expansion definition), A059180.

PierceExp[A_, n_] := Join[Array[1 &, Floor[A]], First@Transpose@ NestList[{Floor[1/Expand[1 - #[[1]] #[[2]]]], Expand[1 - #[[1]] #[[2]]]} &, {Floor[1/(A - Floor[A])], A - Floor[A]}, n - 1]]; PierceExp[N[Log[2], 7!], 25] (* G. C. Greubel, Nov 14 2016 *)

r=1/log(2);for(n=1,30,r=r/(r-floor(r));print1(floor(r),","))

Let u(0)=1/log(2) and u(n+1)=u(n)/frac(u(n)) where frac(x) is the fractional part of x, then a(n)=floor(u(n)).
log(2) = 1/a(1) - 1/(a(1)*a(2)) + 1/(a(1)*a(2)*a(3)) - 1/(a(1)*a(2)*a(3)*a(4)) +- ...
limit n-->infinity a(n)^(1/n) = e.

cp's OEIS Frontend

A091832 Pierce expansion of 1/e^2.

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A091833 Pierce expansion of 1/zeta(2).

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A091846 Pierce expansion of log(2).

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