A184777 -id:A184777 - cp's OEIS frontend

A184774 Primes of the form floor(k*sqrt(2)).

2, 5, 7, 11, 19, 29, 31, 41, 43, 53, 59, 67, 73, 79, 83, 89, 97, 101, 103, 107, 113, 127, 131, 137, 149, 151, 173, 179, 181, 193, 197, 199, 223, 227, 229, 233, 239, 241, 251, 257, 263, 271, 277, 281, 311, 313, 337, 347, 349, 353, 359, 367, 373, 379, 383, 397

Offset: 1

Clark Kimberling, Jan 21 2011

Let N={1,2,...}, L={floor(n*sqrt(2)): n in N} and U={2n+L(n): n in N}. Every prime is in L or U, since the union of the (disjoint) sets L and U is N.

The conjecture formerly posted here, that "if r is an irrational number and 1

Note that every prime not in L is in U={floor(n*s)}, where s=r/(r-1). That is, Beatty sequences partition the primes into two infinite classes.

The conjecture generalized: if r is a positive irrational number and h is a real number, then each of the sets {floor(n*r+h)}, {round(n*r+h)}, and {ceiling(n*r+h)} contains infinitely many primes. Can the method in Vinogradov be extended to cover these cases?

[Update regarding the conjecture from Clark Kimberling, Jan 03 2011.]

			The sequence L(n)=floor(n*sqrt(2)) begins with 1, 2, 4, 5, 7, 8, 9, 11, 12, 14, 15, 16, 18, 19,..., which includes primes L(2)=2, L(4)=5, L(5)=7,...

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A001951 (Beatty sequence of sqrt(2)), A001952 (Beatty sequence of 2+sqrt(2)), A184775, A184776, A184777, A184778, A184779.

[Floor(n*Sqrt(2)): n in [1..400] | IsPrime(Floor(n*Sqrt(2)))]; // Vincenzo Librandi, Apr 30 2015

r=2^(1/2); s=r/(r-1);
a[n_]:=Floor [n*r];  (* A001951 *)
b[n_]:=Floor [n*s];  (* A001952 *)
Table[a[n],{n,1,120}]
t1={}; Do[If[PrimeQ[a[n]], AppendTo[t1,a[n]]], {n,1,600}]; t1
t2={}; Do[If[PrimeQ[a[n]], AppendTo[t2,n]], {n,1,600}]; t2
t3={}; Do[If[MemberQ[t1, Prime[n]], AppendTo[t3,n]],{n,1,300}]; t3
t4={}; Do[If[PrimeQ[b[n]], AppendTo[t4,b[n]]],{n,1,600}]; t4
t5={}; Do[If[PrimeQ[b[n]], AppendTo[t5,n]],{n,1,600}]; t5
t6={}; Do[If[MemberQ[t4, Prime[n]], AppendTo[t6,n]],{n,1,300}]; t6
(* the lists t1,t2,t3,t4,t5,t6 match the sequences
A184774, A184775, A184776, A184777, A184778, A184779 *)
Select[Floor[Range[500]Sqrt[2]],PrimeQ] (* Harvey P. Dale, Jan 05 2019 *)

is(n)=my(k=sqrtint(n^2\2)+1); sqrtint(2*k^2)==n && isprime(n) \\ Charles R Greathouse IV, Apr 29 2015

from math import isqrt
from itertools import count, islice
from sympy import isprime
def A184774_gen(): # generator of terms
    return filter(isprime,(isqrt(k**2<<1) for k in count(1)))
A184774_list = list(islice(A184774_gen(),25)) # Chai Wah Wu, Jul 28 2022

a(n) ~ sqrt(2)*n log n. - Charles R Greathouse IV, Apr 29 2015

A184778 Numbers k such that 2k + floor(k*sqrt(2)) is prime.

Original entry on oeis.org

1, 4, 5, 7, 11, 14, 18, 21, 32, 41, 46, 48, 49, 56, 62, 79, 83, 86, 90, 93, 97, 114, 120, 123, 127, 130, 134, 137, 144, 165, 169, 172, 178, 181, 185, 188, 213, 220, 222, 223, 237, 243, 246, 250, 253, 257, 260, 267, 288, 302, 308, 311, 325, 329, 343, 346, 352, 360, 366, 369, 376

Offset: 1

Clark Kimberling, Jan 21 2011

nonn

			See A184774.

G. C. Greubel, Table of n, a(n) for n = 1..10000

Cf. A184774, A184777, A184779.

r=2^(1/2); s=r/(r-1);
a[n_]:=Floor [n*r];  (* A001951 *)
b[n_]:=Floor [n*s];  (* A001952 *)
Table[a[n],{n,1,120}]
t1={}; Do[If[PrimeQ[a[n]], AppendTo[t1,a[n]]], {n,1,600}]; t1
t2={}; Do[If[PrimeQ[a[n]], AppendTo[t2,n]], {n,1,600}]; t2
t3={}; Do[If[MemberQ[t1, Prime[n]], AppendTo[t3,n]],{n,1,300}]; t3
t4={}; Do[If[PrimeQ[b[n]], AppendTo[t4,b[n]]],{n,1,600}]; t4
t5={}; Do[If[PrimeQ[b[n]], AppendTo[t5,n]],{n,1,600}]; t5
t6={}; Do[If[MemberQ[t4, Prime[n]], AppendTo[t6,n]],{n,1,300}]; t6
(* the lists t1,t2,t3,t4,t5,t6 match the sequences
A184774, A184775, A184776 ,A184777, A184778, A184779 *)

is(n)=isprime(sqrtint(2*n^2)+2*n) \\ Charles R Greathouse IV, May 22 2017

from itertools import count, islice
from math import isqrt
from sympy import isprime
def A184778_gen(): # generator of terms
    return filter(lambda k:isprime((k<<1)+isqrt(k**2<<1)), count(1))
A184778_list = list(islice(A184778_gen(),25)) # Chai Wah Wu, Jul 28 2022

A184775 Numbers k such that floor(k*sqrt(2)) is prime.

Original entry on oeis.org

2, 4, 5, 8, 14, 21, 22, 29, 31, 38, 42, 48, 52, 56, 59, 63, 69, 72, 73, 76, 80, 90, 93, 97, 106, 107, 123, 127, 128, 137, 140, 141, 158, 161, 162, 165, 169, 171, 178, 182, 186, 192, 196, 199, 220, 222, 239, 246, 247, 250, 254, 260, 264, 268, 271, 281, 284, 298, 305, 311, 318

Offset: 1

Clark Kimberling, Jan 21 2011

Chua, Park, & Smith prove a general result that implies that, for any m, there is a constant C(m) such that a(n+m) - a(n) < C(m) infinitely often. - Charles R Greathouse IV, Jul 01 2022

			See A184774.

G. C. Greubel, Table of n, a(n) for n = 1..10000
Lynn Chua, Soohyun Park, and Geoffrey D. Smith, Bounded gaps between primes in special sequences, Proceedings of the AMS, Volume 143, Number 11 (November 2015), pp. 4597-4611. arXiv:1407.1747 [math.NT]

Cf. A001951, A184774, A184776.

r=2^(1/2); s=r/(r-1);
a[n_]:=Floor [n*r];  (* A001951 *)
b[n_]:=Floor [n*s];  (* A001952 *)
Table[a[n],{n,1,120}]
t1={}; Do[If[PrimeQ[a[n]], AppendTo[t1,a[n]]], {n,1,600}]; t1
t2={}; Do[If[PrimeQ[a[n]], AppendTo[t2,n]], {n,1,600}]; t2
t3={}; Do[If[MemberQ[t1, Prime[n]], AppendTo[t3,n]],{n,1,300}]; t3
t4={}; Do[If[PrimeQ[b[n]], AppendTo[t4,b[n]]],{n,1,600}]; t4
t5={}; Do[If[PrimeQ[b[n]], AppendTo[t5,n]],{n,1,600}]; t5
t6={}; Do[If[MemberQ[t4, Prime[n]], AppendTo[t6,n]],{n,1,300}]; t6
(* the lists t1,t2,t3,t4,t5,t6 match the sequences
A184774, A184775, A184776 ,A184777, A184778, A184779 *)

isok(n) = isprime(floor(n*sqrt(2))); \\ Michel Marcus, Apr 10 2018

is(n)=isprime(sqrtint(2*n^2)) \\ Charles R Greathouse IV, Jul 01 2022

from itertools import count, islice
from math import isqrt
from sympy import isprime
def A184775_gen(): # generator of terms
    return filter(lambda k:isprime(isqrt(k**2<<1)), count(1))
A184775_list = list(islice(A184775_gen(),25)) # Chai Wah Wu, Jul 28 2022

A184776 Numbers m such that prime(m) is of the form floor(k*sqrt(2)); complement of A184779.

Original entry on oeis.org

1, 3, 4, 5, 8, 10, 11, 13, 14, 16, 17, 19, 21, 22, 23, 24, 25, 26, 27, 28, 30, 31, 32, 33, 35, 36, 40, 41, 42, 44, 45, 46, 48, 49, 50, 51, 52, 53, 54, 55, 56, 58, 59, 60, 64, 65, 68, 69, 70, 71, 72, 73, 74, 75, 76, 78, 79, 82, 83, 85, 87, 89, 90, 92, 93, 95, 96, 97, 98, 99, 100, 101, 102, 104, 105, 108, 109, 110, 112, 114, 117, 118, 119, 120, 121, 122, 123, 124, 125, 126, 127, 128, 130, 131, 132, 136, 137, 138, 139, 141, 142, 143, 144

Offset: 1

Clark Kimberling, Jan 21 2011

nonn

			See A184774.

G. C. Greubel, Table of n, a(n) for n = 1..10000

Cf. A184774, A184775, A184779.

r=2^(1/2); s=r/(r-1);
a[n_]:=Floor [n*r];  (* A001951 *)
b[n_]:=Floor [n*s];  (* A001952 *)
Table[a[n],{n,1,120}]
t1={}; Do[If[PrimeQ[a[n]], AppendTo[t1,a[n]]], {n,1,600}]; t1
t2={}; Do[If[PrimeQ[a[n]], AppendTo[t2,n]], {n,1,600}]; t2
t3={}; Do[If[MemberQ[t1, Prime[n]], AppendTo[t3,n]],{n,1,300}]; t3
t4={}; Do[If[PrimeQ[b[n]], AppendTo[t4,b[n]]],{n,1,600}]; t4
t5={}; Do[If[PrimeQ[b[n]], AppendTo[t5,n]],{n,1,600}]; t5
t6={}; Do[If[MemberQ[t4, Prime[n]], AppendTo[t6,n]],{n,1,300}]; t6
(* the lists t1,t2,t3,t4,t5,t6 match the sequences
A184774, A184775, A184776 ,A184777, A184778, A184779 *)

from itertools import count, islice
from math import isqrt
from sympy import primepi, isprime
def A184776_gen(): # generator of terms
    return map(primepi,filter(isprime,(isqrt(k**2<<1) for k in count(1))))
A184776_list = list(islice(A184776_gen(),25)) # Chai Wah Wu, Jul 28 2022

A184779 Numbers m such that prime(m) is of the form 2k + floor(k*sqrt(2)); complement of A184776.

Original entry on oeis.org

2, 6, 7, 9, 12, 15, 18, 20, 29, 34, 37, 38, 39, 43, 47, 57, 61, 62, 63, 66, 67, 77, 80, 81, 84, 86, 88, 91, 94, 103, 106, 107, 111, 113, 115, 116, 129, 133, 134, 135, 140, 145, 146, 147, 150, 151, 154, 156, 166, 173, 177, 178, 186, 188, 193, 194, 197, 201, 204, 205, 208

Offset: 1

Clark Kimberling, Jan 21 2011

nonn

			See A184774.

G. C. Greubel, Table of n, a(n) for n = 1..10000

Cf. A184774, A184776.

r=2^(1/2); s=r/(r-1);
a[n_]:=Floor [n*r];  (* A001951 *)
b[n_]:=Floor [n*s];  (* A001952 *)
Table[a[n],{n,1,120}]
t1={}; Do[If[PrimeQ[a[n]], AppendTo[t1,a[n]]], {n,1,600}]; t1
t2={}; Do[If[PrimeQ[a[n]], AppendTo[t2,n]], {n,1,600}]; t2
t3={}; Do[If[MemberQ[t1, Prime[n]], AppendTo[t3,n]],{n,1,300}]; t3
t4={}; Do[If[PrimeQ[b[n]], AppendTo[t4,b[n]]],{n,1,600}]; t4
t5={}; Do[If[PrimeQ[b[n]], AppendTo[t5,n]],{n,1,600}]; t5
t6={}; Do[If[MemberQ[t4, Prime[n]], AppendTo[t6,n]],{n,1,300}]; t6
(* the lists t1,t2,t3,t4,t5,t6 match the sequences
A184774, A184775, A184776 ,A184777, A184778, A184779 *)

from math import isqrt
from itertools import count, islice
from sympy import isprime, primepi
def A184779_gen(): # generator of terms
    return map(primepi,filter(isprime,((k<<1)+isqrt(k**2<<1) for k in count(1))))
A184779_list = list(islice(A184779_gen(),25)) # Chai Wah Wu, Jul 28 2022

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A184774 Primes of the form floor(k*sqrt(2)).

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A184778 Numbers k such that 2k + floor(k*sqrt(2)) is prime.

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A184775 Numbers k such that floor(k*sqrt(2)) is prime.

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A184776 Numbers m such that prime(m) is of the form floor(k*sqrt(2)); complement of A184779.

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A184779 Numbers m such that prime(m) is of the form 2k + floor(k*sqrt(2)); complement of A184776.

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Mathematica

Python