A006784 -id:A006784 - cp's OEIS frontend

A059197 Engel expansion of Pi^e = 22.4592.

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 3, 8, 17, 111, 236, 419, 2475, 3741, 4123, 5563, 5622, 18000, 33641, 42744, 130605, 696987, 975174, 1034590, 2806140, 14026897, 14137435, 65788323, 73121589, 229261119

Offset: 1

Mitch Harris

Cf. A006784 for definition of Engel expansion.

F. Engel, Entwicklung der Zahlen nach Stammbruechen, Verhandlungen der 52. Versammlung deutscher Philologen und Schulmaenner in Marburg, 1913, pp. 190-191.

G. C. Greubel, Table of n, a(n) for n = 1..1022
F. Engel, Entwicklung der Zahlen nach Stammbruechen, Verhandlungen der 52. Versammlung deutscher Philologen und Schulmaenner in Marburg, 1913, pp. 190-191. English translation by Georg Fischer, included with his permission.
P. Erdős and Jeffrey Shallit, New bounds on the length of finite Pierce and Engel series, Sem. Theor. Nombres Bordeaux (2) 3 (1991), no. 1, 43-53.
Index entries for sequences related to Engel expansions

Cf. A059850.

EngelExp[A_, n_] := Join[Array[1 &, Floor[A]], First@Transpose@
NestList[{Ceiling[1/Expand[#[[1]] #[[2]] - 1]], Expand[#[[1]] #[[2]] - 1]/1} &, {Ceiling[1/(A - Floor[A])], (A - Floor[A])/1}, n - 1]];
EngelExp[N[Pi^E, 7!], 100] (* Modified by G. C. Greubel, Dec 28 2016 *)

A059199 Engel expansion of e^gamma (gamma is the Euler-Mascheroni constant A001620) = 1.78107.

Original entry on oeis.org

1, 2, 2, 9, 9, 15, 84, 256, 278, 819, 1734, 6500, 10004, 20116, 26612, 31762827, 181599789, 981641086, 1698644383, 1987894743, 5557385559, 11998593788, 12646182115, 70932754473, 106473857370, 527311590750

Offset: 1

Mitch Harris

Cf. A006784 for definition of Engel expansion

F. Engel, Entwicklung der Zahlen nach Stammbruechen, Verhandlungen der 52. Versammlung deutscher Philologen und Schulmaenner in Marburg, 1913, pp. 190-191.

G. C. Greubel, Table of n, a(n) for n = 1..1000
F. Engel, Entwicklung der Zahlen nach Stammbruechen, Verhandlungen der 52. Versammlung deutscher Philologen und Schulmaenner in Marburg, 1913, pp. 190-191. English translation by Georg Fischer, included with his permission.
P. Erdős and Jeffrey Shallit, New bounds on the length of finite Pierce and Engel series, Sem. Theor. Nombres Bordeaux (2) 3 (1991), no. 1, 43-53.
Index entries for sequences related to Engel expansions

Cf. A073004.

EngelExp[A_, n_] := Join[Array[1 &, Floor[A]], First@Transpose@
NestList[{Ceiling[1/Expand[#[[1]] #[[2]] - 1]], Expand[#[[1]] #[[2]] - 1]/1} &, {Ceiling[1/(A - Floor[A])], (A - Floor[A])/1}, n - 1]];
EngelExp[N[E^EulerGamma, 7!], 100] (* Modified by G. C. Greubel, Dec 28 2016 *)

A067913 Engel expansion of Riemann zeta(5)=sum(i>0,1/i^5).

Original entry on oeis.org

1, 28, 30, 52, 231, 277, 523, 2278, 22749, 48854, 371305, 1447522, 1726931, 1947729, 3657998, 6964377, 14393316, 101963994, 237690542, 487815056, 1525407389, 2636206007, 14146907176, 2348836787752, 5367069498863, 1710637374747486

Offset: 1

Benoit Cloitre, Mar 03 2002

See A006784 for explanation of Engel expansions.

A067921 Engel expansion of sqrt(Pi/2).

Original entry on oeis.org

1, 4, 76, 134, 213, 1649, 1955, 2041, 32363, 217167, 760577, 1633080, 6412486, 24932290, 25544312, 376841489, 426956719, 472747939, 765965856, 2708004694, 5814287795, 14630348817, 21857959576, 92077240148, 184486528542

Offset: 1

Benoit Cloitre, Mar 03 2002

G. C. Greubel, Table of n, a(n) for n = 1..1000

See A006784 for explanation of Engel expansions.

EngelExp[A_, n_] := Join[Array[1 &, Floor[A]], First@Transpose@
NestList[{Ceiling[1/Expand[#[[1]] #[[2]] - 1]], Expand[#[[1]] #[[2]] - 1]} &, {Ceiling[1/(A - Floor[A])], A - Floor[A]}, n - 1]]; EngelExp[N[Sqrt[Pi/2], 7!], 50] (* G. C. Greubel, Jan 12 2017 *)

s=sqrt(asin(1)); for(i=1,30,s=s*ceil(1/s)-1; print1(ceil(1/s),","); );

A068388 Engel expansion of sqrt(3/2).

Original entry on oeis.org

1, 5, 9, 9, 47, 54, 171, 867, 3056, 28687, 133134, 542005, 563497, 1046686, 1955619, 2057281, 42760619, 661780137, 1109113993, 6460565976, 8523453296, 34406061218, 64402180149, 1607033374515, 10943963720662, 124655149151970

Offset: 1

Benoit Cloitre, Mar 03 2002

nonn

Cf. A115754, A382713

Cf. A006784.

print1(1, ", "); s=(3/2)^(1/2); for(i=0,30,s=s*ceil(1/s)-1; print1(ceil(1/s),", "));

A091832 Pierce expansion of 1/e^2.

Original entry on oeis.org

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.

A137507 a(n) = 100^[n/10] + 2n10^[n/10]: inspired by Engel expansion of Pi.

Original entry on oeis.org

1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 300, 320, 340, 360, 380, 400, 420, 440, 460, 480, 14000, 14200, 14400, 14600, 14800, 15000, 15200, 15400, 15600, 15800, 1060000, 1062000, 1064000, 1066000, 1068000, 1070000, 1072000, 1074000, 1076000, 1078000, 100800000, 100820000

Offset: 0

Alexander R. Povolotsky, Apr 23 2008

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,0,0,0,0,120,0,0,0,0,0,0,0,0,0,-2100,0,0,0,0,0,0,0,0,0,10000).

Cf. A006784.

Table[100^(Floor[n/10]) + 2*n*10^(Floor[n/10]), {n,0,50}] (* G. C. Greubel, Feb 23 2017 *)

PARI
```
a(n) = (n + 10^(floor(n/10)))^2 - n^2;
```

vector(30,n, n--; 2*n*10^(n\10) + 100^(n\10)) \\ M. F. Hasler, May 02 2008

[(n + 10^(floor(n/10)))^2 - n^2 for n in range(0,42)] # Stefano Spezia, Apr 16 2025

Edited by M. F. Hasler and N. J. A. Sloane, May 02 2008

A232326 Pierce expansion of 1 to the base Pi.

Original entry on oeis.org

3, 69, 310, 1017, 36745, 214369, 966652, 11159821, 74039764, 550021544, 4481549430, 16543857917, 87205978613, 476981856953, 30989048525367, 203786458494160, 711639924282497, 3174772986229899, 29814569078896025, 100158574806804154

Offset: 0

Peter Bala, Nov 26 2013

Let r and b be positive real numbers. We define a Pierce expansion of r to the base b to be a (possibly infinite) increasing sequence of positive integers [a(0), a(1), a(2), ...] such that we have the alternating series representation r = b/a(0) - b^2/(a(0)*a(1)) + b^3/(a(0)*a(1)*a(2)) - .... Depending on the values of r and b such an expansion may not exist, and if it does exist it may not be unique. When b = 1 and 0 < r < 1 we recover the ordinary Pierce expansion of r.
See A058635, A192223 and A230600 for some predictable Pierce expansions to a base b other than 1.
In the particular case that the base b >= 1 and 0 < r < b then we can find a Pierce expansion of r to the base b as follows:
Define the map f(x) (which depends on the base b) by f(x) = x/b*ceiling(b/x) - 1 and let f^(n)(x) denote the n-th iterate of the map f(x), with the convention that f^(0)(x) = x.
For n = 0,1,2,... define a(n) = ceiling(b/f^(n)(-r)) until f^n(-r) = 0.
Then it can be shown that the sequence of positive integers |a(n)| is a Pierce expansion of r to the base b.
For the present sequence we apply this algorithm with r := 1 and with base b := Pi. See A232325 for an Engel expansion of 1 to the base Pi.

Eric Weisstein's World of Mathematics, Pierce Expansion

Cf. A014014, A006784, A058635, A061233, A192223, A230600, A232325, A232327, A232328.

# Define the n-th iterate of the map f(x) = x/b*ceiling(b/x) - 1
map_iterate := proc(n,b,x) option remember;
if n = 0 then
   x
else
   -1 + 1/b*thisproc(n-1,b,x)*ceil(b/thisproc(n-1,b,x))
end if
end proc:
# Define the (signed) terms of the expansion of x to the base b
a := n -> ceil(evalf(b/map_iterate(n,b,x))):
Digits:= 500:
# Choose values for x and b
x := -1: b:= Pi:
seq(abs(a(n)), n = 0..19);

a(n) = ceiling(Pi/f^(n)(-1)), where f^(n)(x) denotes the n-th iterate of the map f(x) = x/Pi*ceiling(Pi/x) - 1, with the convention that f^(0)(x) = x.
Pierce series expansion of 1 to the base Pi:
1 = Pi/3 - Pi^2/(3*69) + Pi^3/(3*69*310) - Pi^4/(3*69*310*1017) + ....
The associated power series F(z) := 1 - ( z/3 - z^2/(3*69) + z^3/(3*69*310) - z^4/(3*69*310*1017) + ...) has a zero at z = Pi. Truncating the series F(z) to n terms produces a polynomial F_n(z) with rational coefficients which has a real zero close to Pi.

cp's OEIS Frontend

A059197 Engel expansion of Pi^e = 22.4592.

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A059199 Engel expansion of e^gamma (gamma is the Euler-Mascheroni constant A001620) = 1.78107.

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A067913 Engel expansion of Riemann zeta(5)=sum(i>0,1/i^5).

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A067921 Engel expansion of sqrt(Pi/2).

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A068388 Engel expansion of sqrt(3/2).

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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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A137507 a(n) = 100^[n/10] + 2*n*10^[n/10]: inspired by Engel expansion of Pi.

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A137507 a(n) = 100^[n/10] + 2n10^[n/10]: inspired by Engel expansion of Pi.