A070762 -id:A070762 - cp's OEIS frontend

A064628 a(n) = floor((4/3)^n).

1, 1, 1, 2, 3, 4, 5, 7, 9, 13, 17, 23, 31, 42, 56, 74, 99, 133, 177, 236, 315, 420, 560, 747, 996, 1328, 1771, 2362, 3149, 4199, 5599, 7466, 9954, 13273, 17697, 23596, 31462, 41950, 55933, 74577, 99437, 132583, 176777, 235703, 314271, 419028, 558704

Offset: 0

Labos Elemer, Oct 01 2001

a(n) is the perimeter of a hexaflake (rounded down) after n iterations. The total number of holes = A000420(n) - 1. The total number of irregular polygon holes = A000420(n-1) - 1. The total number of triangle holes = 6*A000420(n-1). - Kival Ngaokrajang, Apr 18 2014

a(n) is composite infinitely often (Forman and Shapiro). More exactly, a(n) is divisible by at least one of 2, 3, 5 infinitely often (Dubickas and Novikas). - Tomohiro Yamada, Apr 15 2017

R. K. Guy, Unsolved Problems in Number Theory, E19.

Harry J. Smith, Table of n, a(n) for n=0,...,400
Arturas Dubickas and Aivaras Novikas, Integer parts of powers of rational numbers, Math. Z. 251 (2005), 635--648, available from the first author's page.
W. Forman and H. N. Shapiro, An arithmetic property of certain rational powers, Comm. Pure. Appl. Math. 20 (1967), 561-573.
Kival Ngaokrajang, Illustration of hexaflake for n = 0..3.
Eric Weisstein's World of Mathematics, Power Floors.
Wikipedia, n-flake.

Cf. A002379, A002380, A060692.
Cf. A094969 - A094500.
Cf. A046038, A070761, A070762, A067905 (Composites and Primes).

A064628:=n->floor(4^n/3^n); seq(A064628(n), n=0..30); # Wesley Ivan Hurt, Apr 19 2014

Table[Floor[(4/3)^n], {n, 0, 30}] (* Robert G. Wilson v *)

{ f=t=1; for (n=0, 400, write("b064628.txt", n, " ", f\t); f*=4; t*=3 ) } \\ Harry J. Smith, Sep 20 2009

def A064628(n): return floor((4/3)^n)
[A064628(n) for n in range(0, 46)] # Stefano Spezia, Oct 13 2024

More terms from Robert G. Wilson v, May 26 2004

OFFSET changed from 1 to 0 by Harry J. Smith, Sep 20 2009

A070759 n for which floor((3/2)^n) is prime.

Original entry on oeis.org

2, 3, 4, 5, 6, 7, 21, 22, 98, 106, 164, 189, 219, 364, 443, 670, 775, 1919, 2204, 2715, 3692, 4228, 4912, 10466, 12300, 14087, 21659, 28170, 55832, 66577, 87309, 87505, 98144, 167512, 259517

Offset: 1

Eric W. Weisstein, May 04 2002

nonn

No more terms through 500000. - Ryan Propper, Dec 28 2008

R. K. Guy, Unsolved Problems in Number Theory, E19

Eric Weisstein's World of Mathematics, Power Floors

Cf. A070758, A067904, A070762.

Indices of primes in A002379.

Do[ If[ PrimeQ[ Floor[(3/2)^n]], Print[n]], {n, 1, 12500}]

One more term from Ralf Stephan, Oct 13 2002
Corrected and extended by Robert G. Wilson v, Jan 15 2003
More terms from Ryan Propper, Jan 25 2008
6 more terms from Ryan Propper, Dec 28 2008

A046038 Numbers k for which [ (4/3)^k ] is composite.

Original entry on oeis.org

5, 8, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 39, 40, 41, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82

Offset: 1

Eric W. Weisstein

nonn

Richard K. Guy, Unsolved Problems in Number Theory, 3rd Edition, Springer, 2004, Section E19, p. 338.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Floor Function.

Cf. A070762.

Select[Range[100],CompositeQ[Floor[(4/3)^#]]&] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Oct 01 2019 *)

A070761 Values of floor((4/3)^n) that are composite.

Original entry on oeis.org

4, 9, 42, 56, 74, 99, 133, 177, 236, 315, 420, 560, 747, 996, 1328, 1771, 2362, 3149, 4199, 5599, 7466, 9954, 13273, 17697, 23596, 31462, 41950, 74577, 99437, 132583, 235703, 314271, 419028, 558704, 744938, 993251, 1324335, 1765780, 2354374

Offset: 1

Eric W. Weisstein, May 04 2002

nonn

R. K. Guy, Unsolved Problems in Number Theory, E19

Eric Weisstein's World of Mathematics, Power Floors

Cf. A046038, A067905, A070762.

Select[Rest[Union[Floor[(4/3)^Range[80]]]],!PrimeQ[#]&] (* Harvey P. Dale, Sep 23 2011 *)

A072924 Least k such that floor((1+1/k)^n) is prime.

Original entry on oeis.org

1, 2, 2, 2, 2, 2, 2, 4, 3, 3, 3, 3, 6, 8, 7, 6, 6, 7, 5, 11, 2, 2, 9, 4, 6, 10, 5, 9, 5, 6, 4, 7, 10, 11, 7, 6, 4, 3, 10, 4, 4, 3, 5, 4, 17, 6, 11, 7, 5, 14, 12, 8, 6, 11, 4, 14, 8, 7, 3, 16, 4, 21, 8, 12, 7, 8, 7, 7, 18, 12, 8, 17, 10, 12, 28, 6, 24, 16, 12, 16, 18, 7, 6, 6, 7, 11, 8, 14, 24, 8

Offset: 1

Benoit Cloitre, Aug 11 2002

a(n) = 2 for n in A070759. a(n) = 3 for n in A070762 but not in A070759. - Robert Israel, Jan 09 2018

R. K. Guy, Unsolved Problems in Number Theory, E19

Robert Israel, Table of n, a(n) for n = 1..10000
Robert Israel, Scatter plot of a(n)/sqrt(n) for n=1..15000

Cf. A070759, A070762

f:= proc(n) local k;
  for k from 1 do if isprime(floor((1+1/k)^n)) then return k fi od
end proc:
map(f, [$1..100]); # Robert Israel, Jan 09 2018

lkp[n_]:=Module[{k=1},While[!PrimeQ[Floor[(1+1/k)^n]],k++];k]; Array[ lkp,90] (* Harvey P. Dale, Dec 02 2018 *)

a(n)=if(n<0,0,s=1; while(isprime(floor((1+1/s)^n)) == 0,s++); s)

It seems that a(n)/sqrt(n) is bounded. More precisely for n large enough it seems that (1/2)*sqrt(n) < a(n) < 3*sqrt(n).

On the contrary, A.L. Whiteman conjectured that the sequence floor(r^n) for non-integer rational r > 1 always contains infinitely many primes. If this conjecture is true for some r=1+1/k, then lim inf_{n -> infinity} a(n) is finite. - Robert Israel, Jan 09 2018

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A064628 a(n) = floor((4/3)^n).

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A070759 n for which floor((3/2)^n) is prime.

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A046038 Numbers k for which [ (4/3)^k ] is composite.

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Mathematica

A070761 Values of floor((4/3)^n) that are composite.

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Programs

Mathematica

A072924 Least k such that floor((1+1/k)^n) is prime.

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Programs

Maple

Mathematica

PARI

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