cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-4 of 4 results.

A076255 a(n) = floor(t^n), where t=3450844193^(1/9) (approximately 11.4754).

Original entry on oeis.org

11, 131, 1511, 17341, 198997, 2283583, 26205133, 300715537, 3450844193, 39599967967, 454427199648, 5214753707584, 59841612147821, 686709046151502, 7880290940381527, 90429834371744720, 1037722465625775937
Offset: 1

Views

Author

David Terr, Nov 05 2002

Keywords

Comments

FEPS(9,1) (first floor exponential prime sequence of length 9).
A floor exponential prime sequence (FEPS) is a sequence of the form {a(n) = floor[t^n]:1<=n<=length} in which t is a real number greater than or equal to 2 and each term in the sequence is prime. FEPS(len,k) is the k-th maximal optimal floor exponential prime sequence of length len, ordered by exponent t = a(len)^(1/len). As far as I know, the only previously known FEPS was FEPS(8,1) = {2, 5, 13, 31, 73, 173, 409, 967} (first 8 terms of A063636). During the past few days I've discovered 20 others with length up to 92, including 16 of length up to 27 which I know to be the first such sequence of given length.
I found that past the first nine members, the only other powers of t which produce a prime are 15, 79 & 101 and no others <= 2500. - Robert G. Wilson v

Examples

			a(4) = floor(t^4) = floor(3450844193^(4/9)) = 17341, which is prime, like each other term in the sequence.

References

  • R. Crandall and C. Pomerance, Prime Numbers: A Computational Perspective, Springer, NY, 2001; see page 69, exercise 1.75.

Crossrefs

Cf. A063636.

Programs

  • Mathematica
    Table[ Floor[3450844193^(n/9)], {n, 1, 18}]

Extensions

Edited and extended by Robert G. Wilson v, Nov 07 2002

A076357 a(n) = floor(t^n) where t = 39661481813^(1/10) (approximately 11.4772).

Original entry on oeis.org

1, 11, 131, 1511, 17351, 199151, 2285711, 26233621, 301089179, 3455668247, 39661481813, 455203748458, 5224475817304, 59962484179977, 688202919252740, 7898659712736578, 90654694294744401, 1040464318828877723, 11941643035453940036, 137056923342374688074
Offset: 0

Views

Author

David Terr, Nov 06 2002

Keywords

Comments

FEPS(10, 1) (the first floor exponential prime sequence of length 10).
See A076255 for more explanation of floor exponential prime sequences.
a(n) is prime for n = 1..10.
I found that past the first ten members, there are no powers of t which produce a prime <= 2000. - Robert G. Wilson v, Nov 08 2002

Examples

			a(5) = floor(t^5) = floor(39661481813^(1/2)) = 199151.

References

  • Richard Crandall and Carl Pomerance, Prime Numbers - a Computational Perspective, Springer, 2001, page 69, exercise 1.75.

Crossrefs

Cf. A063636, A076255.

Programs

  • Mathematica
    Table[ Floor[39661481813^(n/10)], {n, 1, 17}]

Extensions

Edited and extended by Robert G. Wilson v, Nov 08 2002

A086758 a(n) is the smallest m such that the integer part of the first n powers of m^(1/n) are primes.

Original entry on oeis.org

2, 5, 13, 31, 631, 173, 409, 967, 3450844193, 39661481813, 2076849234433, 52134281654579, 14838980942616539, 260230524377962793, 4563650703502319197, 80032531899785490253, 172111744128569095516889
Offset: 1

Views

Author

Farideh Firoozbakht, Aug 01 2003

Keywords

Comments

All terms of this sequence must be primes because floor((a(n)^(1/n))^n) = a(n).
Floor[(a(8)^(1/8))^k] = floor[(1287/545)^k] for k=1..10 (see puzzle 227). If a(9) exists it must be greater than 22000000.

Examples

			a(5)=631 because floor(631^(1/5)) = 3, floor(631^(2/5)) = 13, floor(631^(3/5)) = 47, floor(631^(4/5)) = 173 and floor(631^(5/5)) = 631 are primes and 631 is the smallest m with this property.
a(8)=967 because the sequence {2, 5, 13, 31, 73, 173, 409, 967} consists entirely of primes, the i-th term in the sequence being floor(967^(i/8)) and 967 is the smallest integer with this property.

References

  • R. Crandall and C. Pomerance, Prime Numbers: A Computational Perspective, Springer, NY, 2001; see Exercise 1.75, p. 69.

Links

Crossrefs

Cf. A063636, A076255, A076357.

Programs

  • Mathematica
    Do[Print[For[m=1, Union[Table[PrimeQ[Floor[Prime[m]^(k/n)]], {k, n}]]!={True}, m++ ]; Prime[m]], {n, 8}]

Formula

For[m=1, Union[Table[PrimeQ[Floor[Prime[m]^(k/n)]], {k, n}]]!={True}, m++ ]; Prime[m]

Extensions

Terms a(9) and following from Jon E. Schoenfield, May 15 2010

A094106 a(n) is the maximal length L of a "power floor prime" sequence, i.e., a sequence of the form floor(x^k), k = 1, 2, ..., L such that floor(x) = prime(n).

Original entry on oeis.org

8, 7, 8, 5, 10, 12, 16, 14, 18, 22, 24, 26, 27, 28, 34, 35, 37, 39, 40, 45, 43, 46, 49, 51, 55, 57
Offset: 1

Views

Author

Johann Wiesenbauer (j.wiesenbauer(AT)tuwien.ac.at), May 02 2004

Keywords

Examples

			a(1)=8 because for x=111/47 the sequence [x^k], k=1,2,... 2,5,13,31,73,173,409,967,... starts with 8 primes and this is the maximum for any x with [x]=2. (Compare also A063636, though the rational number x = 1287/545 used there is not of minimal height!)

References

  • Crandall and Pomerance, "Prime numbers, a computational perspective", p. 69, Research Problem 1.75.

Links

Crossrefs

Cf. A076255, A076357.

Extensions

a(22) = 46 from Johann Wiesenbauer (j.wiesenbauer(AT)tuwien.ac.at), Jun 03 2004
a(23) = 49 from Johann Wiesenbauer (j.wiesenbauer(AT)tuwien.ac.at), Jun 27 2004
a(24) = 51 from Johann Wiesenbauer (j.wiesenbauer(AT)tuwien.ac.at), Aug 08 2004
a(25) and a(26) from Michael Kenn (michael.kenn(AT)philips.com), Jan 03 2006, who says: To achieve this result I used a shared network of 37 computers over the Christmas holidays. The total calculation time was equivalent to slightly more than 1 CPU year of a P4 - 2.4GHz.
Showing 1-4 of 4 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.