A002581 -id:A002581 - cp's OEIS frontend

A002946 Continued fraction for cube root of 3.

1, 2, 3, 1, 4, 1, 5, 1, 1, 6, 2, 5, 8, 3, 3, 4, 2, 6, 4, 4, 1, 3, 2, 3, 4, 1, 4, 9, 1, 8, 4, 3, 1, 3, 2, 6, 1, 6, 1, 3, 1, 1, 1, 1, 12, 3, 1, 3, 1, 1, 4, 1, 6, 1, 5, 1, 2, 1, 3, 3, 11, 8, 1, 139, 8, 2, 8, 5, 1, 2, 2, 2, 2, 3, 1, 1, 2, 1, 1, 1, 52, 2, 46, 2, 2, 3

Offset: 0

N. J. A. Sloane

			3^(1/3) = 1.44224957030740838... = 1 + 1/(2 + 1/(3 + 1/(1 + 1/(4 + ...)))). - _Harry J. Smith_, May 08 2009

H. P. Robinson, Letter to N. J. A. Sloane, Nov 13 1973.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Harry J. Smith, Table of n, a(n) for n = 0..19999
S. Lang and H. Trotter, Continued fractions for some algebraic numbers, J. Reine Angew. Math. 255 (1972), 112-134.
S. Lang and H. Trotter, Continued fractions for some algebraic numbers, J. Reine Angew. Math. 255 (1972), 112-134. [Annotated scanned copy]
Herman P. Robinson, Letter to N. J. A. Sloane, Nov 13 1973.
G. Xiao, Contfrac
Index entries for continued fractions for constants

Cf. A002581 (decimal expansion).

Cf. A002353, A002354 (convergents).

SetDefaultRealField(RealField(100)); ContinuedFraction(3^(1/3)); // G. C. Greubel, Nov 02 2018

with(numtheory): cfrac(3^(1/3),80,'quotients'); # Muniru A Asiru, Nov 02 2018

ContinuedFraction[Power[3, (3)^-1],120] (* Harvey P. Dale, May 11 2011 *)

{ allocatemem(932245000); default(realprecision, 21000); x=contfrac(3^(1/3)); for (n=1, 20000, write("b002946.txt", n-1, " ", x[n])); } \\ Harry J. Smith, May 08 2009

Offset changed by Andrew Howroyd, Jul 04 2024

A243406 Decimal expansion of 8^(1/sqrt(8)).

Original entry on oeis.org

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

Offset: 1

Stanislav Sykora, Jun 05 2014

			2.0858857967599065528890408701429704566560332036958385127160...

Stanislav Sykora, Table of n, a(n) for n = 1..2000

Cf. A002581, A243443 (k=7), A243444 (k=6).

RealDigits[8^(1/Sqrt[8]),10,120][[1]] (* Harvey P. Dale, Nov 05 2019 *)

default(realprecision,20080);8.0^(1.0/sqrt(8.0));

A243443 Decimal expansion of 7^(1/sqrt(7)).

Original entry on oeis.org

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

Offset: 1

Stanislav Sykora, Jun 05 2014

The largest k^(1/sqrt(k)), for any natural number k, which occurs for k = 7 = A000227(2).

			2.086493496309364423191012078331874644759917871182477044311483403077176...

Stanislav Sykora, Table of n, a(n) for n = 1..2000

Cf. A002581, A243406 (k=8), A243444 (k=6).

RealDigits[7^(1/Sqrt[7]),10,120][[1]] (* Harvey P. Dale, Aug 17 2024 *)

default(realprecision, 20080); 7.0^(1.0/sqrt(7.0));

A166928 Decimal expansion of smaller solution to 3^x = x^3.

Original entry on oeis.org

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

Offset: 1

Franklin T. Adams-Watters, Oct 23 2009

The larger solution is of course 3.
Also, the limit of infinite tetration a^a^...^a of a=3^(1/3) (=A002581), i.e., lim_{n->oo} x(n) where x(n+1)=a^x(n), x(1)=a. - M. F. Hasler, Nov 03 2013
The constant is transcendental (Vassilev-Missana, p. 23). - Peter Bala, Jan 01 2014

			2.47805268028830241189373651689...

M. Vassilev-Missana, Some results on infinite power towers, Notes on Number Theory and Discrete Mathematics, Vol. 16 (2010) No. 3, 18-24.
Index entries for transcendental numbers

Cf. A073084, A341328.

RealDigits[ -3*ProductLog[ -Log[3]/3 ] / Log[3], 10, 105] // First (* Jean-François Alcover, Mar 05 2013 *)

PARI
```
solve(x=2,exp(1),3^x-x^3)
```

A210706 Numbers k such that floor[ 3^(1/3)*10^k ] is prime.

Original entry on oeis.org

23, 80, 2487

Offset: 1

M. F. Hasler, Aug 31 2013

Inspired by prime curios about 4957 (cf. LINKS), one of which honors the late Bruce Murray (Nov 30 1931 - Aug 29 2013).
Meant to be a "condensed" version of A210704, see there for more.
Alternate definition: Numbers k such that concatenation of the first (k+1) digits of A002581 yields a prime.

			t = 3^(1/3) = 1.44224957030740838232163831... multiplied by 10^23 yields
t*10^23 = 144224957030740838232163.831..., the integer part of which is the prime A210704(1), therefore a(1)=23.

C. Caldwell, G. L. Honaker (Eds.), 4957 (another Prime Pages' Curiosity).

Cf. A002581 = decimal expansion of 3^(1/3).

Cf. A065815 (analog for gamma), A060421 (1+ analog for Pi), A064118 (1+ analog for exp(1)), A119344 (1 + analog for sqrt(3)), A136583 (1+ analog for sqrt(10)).

\p2999
t=sqrtn(3,3);for(i=1,2999,ispseudoprime(t\.1^i)&print1(i","))

a(n) = (length of A210704(n)) - 1, where "length" means number of decimal digits.

A243405 Minimum among the numbers p^(n/p), where p is a prime factor of n.

Original entry on oeis.org

1, 2, 3, 4, 5, 8, 7, 16, 27, 25, 11, 64, 13, 49, 125, 256, 17, 512, 19, 625, 343, 121, 23, 4096, 3125, 169, 19683, 2401, 29, 15625, 31, 65536, 1331, 289, 16807, 262144, 37, 361, 2197, 390625, 41, 117649, 43, 14641, 1953125, 529, 47, 16777216, 823543, 9765625, 4913, 28561, 53

Offset: 1

Stanislav Sykora, Jun 04 2014

nonn

The setting a(1)=1 is conventional.

Upper bound (for any n): a(n) <= (3^(1/3))^n = A002581^n.

			a(12)=64 because 2^(12/2)=64 is smaller than 3^(12/3)=81.

Stanislav Sykora, Table of n, a(n) for n = 1..2000

Cf. A002581, A092975 (maximum instead of minimum), A033845.

A243405(n)= {my(m,k,p,q);if(n==1,return(1));
  p=factor(n);m=2^n;
  for(k=1,#p[,1],q=p[k,1]^(n\p[k,1]);if(q

For prime p, a(p)=p.
For n>1: When gpf(n)>3 then a(n)=gpf(n)^(n/gpf(n)); otherwise if n is even then a(n)=2^(n/2); otherwise a(n)=3^(n/3).
If n is in A033845, a(n) = 2^(n/2); otherwise a(n) = gpf(n)^(n/gpf(n)). - Franklin T. Adams-Watters, Jun 15 2014

A319034 Decimal expansion of the height that minimizes the total surface area of the four triangular faces of a square pyramid of unit volume.

Original entry on oeis.org

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

Offset: 1

Jon E. Schoenfield, Oct 22 2018

A square pyramid with a height of h and a base of size s X s has volume V = (1/3)*s^2*h, so a square pyramid of unit volume has s = sqrt(3/h), and the slant height of each of the four triangular faces is t = sqrt(h^2 + (s/2)^2) = sqrt(h^2 + 3/(4*h)), and the total area of the four faces is A = 4*(s*t/2) = sqrt(12*h^3 + 9)/h; this area is minimized at h = (3/2)^(1/3), where it reaches A = 3^(7/6)*2^(1/3).

If the total surface area of all five faces including the square base is to be minimized, then the resulting height is 6^(1/3) (cf. A005486). - Jon E. Schoenfield, Nov 11 2018

			1.14471424255333186780804221193967700891590692078793...

Eric Weisstein's World of Mathematics, Pyramid.

Cf. A002580, A002581, A005486, A010584.

RealDigits[Surd[3/2, 3], 10, 120][[1]] (* Amiram Eldar, Jun 21 2023 *)

sqrtn(3/2, 3) \\ Michel Marcus, Oct 23 2018

Equals (3/2)^(1/3) = (1/2)*A010584.

Equals A002581/A002580. - Michel Marcus, Oct 23 2018

A010648 Decimal expansion of cube root of 78.

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 1..5000

RealDigits[N[78^(1/3), 100]][[1]] (* Vincenzo Librandi, Mar 29 2013 *)

sqrtn(78,3) \\ Charles R Greathouse IV, Apr 14 2014

Equals A005486*A010585 = A002581*A010598. [Bruno Berselli, Mar 29 2013]

A210973 Decimal expansion of cube root of (3/4).

Original entry on oeis.org

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

Offset: 0

Omar E. Pol, Aug 09 2012

Radius of a sphere with volume Pi.

			0.908560296416069829445605878... =  A002581 / A005480.

Chai Wah Wu, Table of n, a(n) for n = 0..10000

Cube root of A152627.

Cf. A005486.

root[3](3/4) ;evalf(%) ; # R. J. Mathar, Sep 14 2012

RealDigits[Surd[3/4,3],10,120][[1]] (* Harvey P. Dale, Sep 27 2015 *)

sqrtn(3/4,3) \\ Charles R Greathouse IV, Apr 14 2014

(3/4)^(1/3).

A005486/2.

A251735 Decimal expansion of Sum_{n>=1} (-1)^(n+1)/n^(1/3).

Original entry on oeis.org

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

Offset: 0

R. J. Mathar, Dec 07 2014

Cubic root analog of A113024.

			0.57175283382527766493...

G. C. Greubel, Table of n, a(n) for n = 0..10000

Cf. A005480, A113024, A251734.

Maple
```
Zeta(1/3)*(1-root[3](4)) ; evalf(%) ;
```

RealDigits[-Zeta[1/3]*(4^(1/3) - 1), 10, 100][[1]] (* G. C. Greubel, Apr 15 2018 *)

-zeta(1/3)*(4^(1/3)-1) \\ Charles R Greathouse IV, Apr 20 2016

Equals 1 - 1/A002580 + 1/A002581 - 1/A005480 + ... = A251734 *(1 - A005480).

cp's OEIS Frontend

A002946 Continued fraction for cube root of 3.

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A243406 Decimal expansion of 8^(1/sqrt(8)).

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A243443 Decimal expansion of 7^(1/sqrt(7)).

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A166928 Decimal expansion of smaller solution to 3^x = x^3.

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A210706 Numbers k such that floor[ 3^(1/3)*10^k ] is prime.

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A243405 Minimum among the numbers p^(n/p), where p is a prime factor of n.

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A319034 Decimal expansion of the height that minimizes the total surface area of the four triangular faces of a square pyramid of unit volume.

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