A060699 -id:A060699 - cp's OEIS frontend

A063636 a(n) = floor((1287/545)^n).

2, 5, 13, 31, 73, 173, 409, 967, 2283, 5392, 12735, 30073, 71017, 167706, 396032, 935217, 2208486, 5215270, 12315692, 29083113, 68678837, 162182870, 382989640, 904417737, 2135753445, 5043513182, 11910094433, 28125305569, 66417005997

Offset: 1

Jud McCranie, Aug 10 2001

nonn

The first eight terms are primes. Does there exist a number theta such that the floor of theta^n is always prime?

			(1287/545)^3 = 13.16879..., so a(3)=13.

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

Harry J. Smith, Table of n, a(n) for n = 1..300

Cf. A000040, A051254, A060449, A060699.

{ for (n=1, 300, write("b063636.txt", n, " ", 1287^n \ 545^n); ) } \\ Harry J. Smith, Aug 26 2009

A060449 Generalized Mills numbers: a(n) = floor(c^(b^n)) where c=4.4, b=1.179.

Original entry on oeis.org

5, 7, 11, 17, 29, 53, 109, 252, 679, 2184, 8650, 43828, 296913, 2832896, 40474353, 930818760, 37522518949, 2931502379404, 499688559138590, 213681340556825047, 270268120176240462240, 1227682581046707804164120

Offset: 1

Jason Earls, Apr 07 2001

First seven terms are primes.

Harry J. Smith, Table of n, a(n) for n = 1..44
C. Rivera, Prime Puzzles

Cf. A051254, A108739, A051021, A060699, A191357.

Digits := 100; A060449 := n->4.4^(1.179^n);

{ default(realprecision, 2000); for (n=1, 44, write("b060449.txt", n, " ", floor(4.4^(1.179^n))); ) } \\ Harry J. Smith, Jul 05 2009

More terms from James Sellers, Apr 11 2001

Offset changed from 0 to 1 by Harry J. Smith, Jul 05 2009

A243358 The densest possibly infinite sequence of primes of the form a(n) = floor[A^(C^n)] for A < 2. The density parameter C here approaches its minimal possible value C_0 = 1.2209864... (A117739), while the corresponding value of A is 1.8252076... (A243370).

Original entry on oeis.org

2, 2, 2, 3, 5, 7, 11, 19, 37, 83, 223, 739, 3181, 18911, 166657, 2375617, 60916697, 3199316947, 403223394631, 147983594957101, 200280265936061027, 1333721075205083093951, 62146579709944366260614273, 31146685223026045243771057244741

Offset: 1

Andrey V. Kulsha, Jun 03 2014

nonn

Double-checked by David J. Broadhurst. Terms from a(61) to a(67) from David J. Broadhurst. Terms after a(52) are strong probable primes.

It is very likely, but not yet proved, that the sequence is infinite. However, it is clear that for density parameter C < C_0 = 1.2209864... (see A117739) such a sequence must contain nonprime terms.

Andrey V. Kulsha, Table of n, a(n) for n = 1..40
Andrey V. Kulsha and David J. Broadhurst, Table of n, a(n) for n = 1..67
Chris K. Caldwell, A proof of a generalization of Mills' Theorem

Cf. A060699, A117739, A243370.

Once the terms up to the prime 223 are known, the following algorithm works:
1. assign P:=(the largest prime currently in the sequence)
2. assign k:=(the distance between 83 and P in the sequence)
3. assign C:=(logP/log84)^(1/k)
4. assign P:=P^C
5. if floor[P] is prime, add it to the sequence and go to 4
6. add nextprime[P] to the sequence and go to 1
That algorithm gives heuristically as many terms as needed because the increment of C at step 3 becomes so tiny that the values of 84^(C^n) for n < k don't jump over integers anymore (although there's no proof).
So we have a(n) = floor[(84-0)^(C_0^(n-10))], where C_0 = 1.2209864... (see A117739), and "84-0" notation means that when C approaches C_0 from above, the necessary value of A brings A^(C^10) to 84 from below.

A191357 Floor(A^(C^n)), where A = 32.76 and C = 1.33.

Original entry on oeis.org

103, 479, 3673, 55147, 2024063, 243937297, 142915724779, 685893080269745, 53978528420922581864, 175329092084368391071206608, 80227969100540338877503013472650510, 26469961649988241699181245714190498215773679043

Offset: 1

Arkadiusz Wesolowski, May 31 2011

nonn

First seven terms are primes.

			a(2) = 479 because 32.76^(1.33^2) = 479.1724192479....

Chris Caldwell, A proof of a generalization of Mills' Theorem
G. L. Honaker, Jr. and Chris Caldwell, Prime Curios! 142915724779
Carlos Rivera, Puzzle 85
Eric Weisstein's World of Mathematics, Floor Function

Cf. A051254, A108739, A051021, A060449, A060699.

default(realprecision, 100); for(n=1, 12, print1(floor(32.76^(1.33^n)), ", ")); \\ Arkadiusz Wesolowski, Jul 18 2011

a(n) = floor(32.76^(1.33^n)).

cp's OEIS Frontend

A063636 a(n) = floor((1287/545)^n).

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A060449 Generalized Mills numbers: a(n) = floor(c^(b^n)) where c=4.4, b=1.179.

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A243358 The densest possibly infinite sequence of primes of the form a(n) = floor[A^(C^n)] for A < 2. The density parameter C here approaches its minimal possible value C_0 = 1.2209864... (A117739), while the corresponding value of A is 1.8252076... (A243370).

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A191357 Floor(A^(C^n)), where A = 32.76 and C = 1.33.

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