A184774 -id:A184774 - cp's OEIS frontend

A184796 Primes of the form floor(k*sqrt(3)).

3, 5, 13, 17, 19, 29, 31, 41, 43, 53, 67, 71, 79, 83, 103, 107, 109, 131, 157, 173, 181, 193, 197, 199, 211, 223, 233, 239, 251, 263, 271, 277, 311, 313, 337, 349, 353, 367, 379, 389, 401, 419, 431, 433, 439, 443, 457, 467, 479, 491, 509, 521, 523, 547, 557, 569, 571, 587, 599, 601, 607, 613, 647, 659, 661, 673, 677, 691, 701, 727, 739, 743, 751, 769, 827, 829, 853, 857, 859, 881, 883, 907, 911, 919, 937, 947, 971, 983, 997, 1009, 1013, 1021, 1039

Offset: 1

Clark Kimberling, Jan 22 2011

nonn

See A184774.

Equals the prime terms of A022838. - Bill McEachen, Oct 28 2021

			The sequence A022838(n)=floor(n*sqrt(3)) begins with 1,3,5,6,8,10,12,13,15,17,19,... which includes the primes A022838(2)=3, A022838(3)=5, A022838(8)=13,...

Cf. A184774, A184797, A184798, A184799, A184800, A184801.

Cf. A022838. - Bill McEachen, Oct 28 2021

r=3^(1/2); s=r/(r-1);
a[n_]:=Floor [n*r];  (* A022838 *)
b[n_]:=Floor [n*s];  (* A054406 *)
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
A184796, A184797, A184798, A184799, A184800, A184801. *)

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

A184777 Primes of the form 2k + floor(k*sqrt(2)).

Original entry on oeis.org

3, 13, 17, 23, 37, 47, 61, 71, 109, 139, 157, 163, 167, 191, 211, 269, 283, 293, 307, 317, 331, 389, 409, 419, 433, 443, 457, 467, 491, 563, 577, 587, 607, 617, 631, 641, 727, 751, 757, 761, 809, 829, 839, 853, 863, 877, 887, 911, 983, 1031, 1051, 1061, 1109, 1123, 1171, 1181, 1201, 1229, 1249, 1259, 1283, 1297, 1307, 1321, 1399, 1423, 1427, 1433, 1447, 1451, 1471, 1481, 1543, 1553, 1567, 1597, 1601, 1621, 1669, 1693, 1741, 1789, 1823, 1847, 1867, 1877, 1901

Offset: 1

Clark Kimberling, Jan 21 2011

nonn

			See A184774.

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

Cf. A184774, A184778, 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 *)
Select[Table[2k+Floor[k Sqrt[2]],{k,1000}],PrimeQ] (* Harvey P. Dale, Mar 06 2025 *)

from math import isqrt
from itertools import count, islice
from sympy import isprime
def A184777_gen(): # generator of terms
    return filter(isprime,((k<<1)+isqrt(k**2<<1) for k in count(1)))
A184777_list = list(islice(A184777_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

A184802 Primes of the form floor(k*sqrt(5)).

Original entry on oeis.org

2, 11, 13, 17, 29, 31, 53, 67, 71, 73, 89, 107, 109, 127, 131, 149, 163, 167, 181, 199, 223, 239, 241, 257, 263, 277, 281, 283, 313, 317, 337, 353, 373, 389, 409, 431, 433, 449, 467, 487, 491, 503, 509, 521, 523, 541, 547, 563, 599, 601, 617, 619, 641, 643

Offset: 1

Clark Kimberling, Jan 22 2011

nonn

See A184774.

			The sequence U(n)=floor(n*sqrt(5)) begins with
2,4,6,8,11,13,15,17,20,22,24,26,29,...,
which includes the primes U(1)=2, U(5)=11,...

Cf. A184774, A022839, A108598, A184802, A184803, A184804, A184805, A184806, A184807.

r=5^(1/2); s=r/(r-1);
a[n_]:=Floor [n*r];  (* A022839 *)
b[n_]:=Floor [n*s];  (* A108598 *)
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
A184802, A184803, A184804, A184805, A184806, A184807. *)

for(k=1,300,isprime(p=sqrtint(k^2*5))&&print1(p",")) \\ M. F. Hasler, Aug 26 2014

A184792 Numbers k such that floor(k*r) is prime, where r = golden ratio=(1+sqrt(5))/2.

Original entry on oeis.org

2, 7, 11, 12, 18, 23, 27, 33, 37, 38, 42, 44, 49, 60, 63, 64, 70, 79, 81, 85, 86, 101, 107, 111, 112, 122, 123, 131, 138, 142, 148, 149, 159, 163, 168, 174, 175, 190, 194, 196, 205, 215, 216, 222, 227, 231, 237, 241, 248, 253, 259, 268, 274, 278, 283, 285, 289, 301, 304, 309, 311, 315, 322, 348, 352, 353, 357, 363, 367, 372, 379, 383, 390, 398, 400, 404, 409, 416, 419, 457, 468, 478, 487, 493, 500, 508, 509, 519, 530, 531, 545, 546, 561, 568, 582, 589, 598

Offset: 1

Clark Kimberling, Jan 22 2011

nonn

			The sequence L(n)=floor(n*r) begins with
1,3,4,6,8,9,11,12,14,16,17,...,
which includes the primes L(2)=3, L(7)=11,...

Cf. A184774, A095280, A184792, A184793, A095281, A184794, A184795.

r=(1+5^(1/2))/2; s=r/(r-1);
a[n_]:=Floor [n*r];  (* A095280 *)
b[n_]:=Floor [n*s];  (* A095281 *)
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
A095280, A184792, A184793, A095281, A184794, A184795 *)
Select[Range[600],PrimeQ[Floor[GoldenRatio #]]&] (* Harvey P. Dale, Mar 28 2024 *)

A184859 Primes of the form floor(kr+h), where r=(1+sqrt(5))/2 and h=1/2.

Original entry on oeis.org

2, 3, 5, 11, 13, 19, 23, 29, 31, 37, 47, 53, 61, 71, 73, 79, 83, 89, 97, 107, 113, 131, 139, 149, 157, 163, 167, 173, 181, 191, 193, 197, 199, 223, 227, 233, 239, 241, 251, 257, 269, 277, 283, 293, 307, 311, 317, 337, 349, 353, 359, 367, 379, 383, 401, 409, 419, 421, 443, 461, 463, 479, 487, 503, 521, 523, 547, 557, 563, 571, 587, 599, 607, 613, 631, 641, 647, 659, 673, 683, 691, 701, 709, 733, 739, 743, 751, 757, 769, 773, 809, 811, 827, 853, 859, 877, 883, 887, 911, 919, 929, 937, 947, 953, 971

Offset: 1

Clark Kimberling, Jan 23 2011

nonn

See "conjecture generalized" at A184774.

			The sequence U(n)=floor(n*r+h) begins with
2,3,5,6,8,10,11,13,15,16,18,19,...,
which includes the primes U(1)=2, U(2)=3,...

Cf. A184774, A184859, A184860, A184861.

r=(1+5^(1/2))/2; h=1/2;s=r/(r-1);
a[n_]:=Floor [n*r+h];
Table[a[n],{n,1,120}]  (* A007067 *)
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
(* Lists t1, t2, t3 match A184859, A184860, A184861. *)
Select[Floor[GoldenRatio*Range[600]+1/2],PrimeQ] (* Harvey P. Dale, Jan 02 2013 *)

A228639 Decimal expansion of Sum_{n>=1} (-1)^floor(n*sqrt(2))/n.

Original entry on oeis.org

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

Offset: 0

Vaclav Kotesovec, Aug 31 2013

From Jon E. Schoenfield, Jul 08 2015: (Start)
If we define the partial sum s_n = Sum_{i=1..n} (-1)^floor(i*sqrt(2))/i then the real-valued sequence s_1, s_2, s_3, ... converges very slowly, and the convergence is not smooth because of the aperiodicity created by the (-1)^floor(i*sqrt(2)) factor. However, if we define the partial sum S_j = s_(n_j) where n_j is the j-th Pell number, then the real-valued sequence S_1, S_2, S_3, ... converges fairly quickly. It appears that, for either even or odd values of j, as j increases, S_j approaches
     c0 + (c1 + k1*log(n))/n_j + (c2 + k2*log(n))/(n_j)^2 + (c3 + k3*log(n))/(n_j)^3 + ...,
but the coefficients c1, k1, c2, k2, c3, k3, etc. take one set of values when j is even and a different set of values when j is odd.
However, it also appears that, if we define the sequence of real numbers t_1, t_2, ... that results from adjusting each of the terms of the S_j sequence using
    t_j = S_j - Sum{i=1..D} (-1)^(ij)*sqrt(2)*(j*d_i+(j mod 2)) / r^(j*(2i-1))
where r is the limit of the ratio Pell(j+1)/Pell(j) as j increases, i.e., r = 1 + sqrt(2), and d is the (empirically-determined) integer sequence
     d = {0, 1, 3, 61, 395, 47041, 504987, 182501677, 2705354787, 2186736573121, ...}
(from which we use the first D terms in the above formula for t_j), then the sequence of real numbers t_1, t_2, ... follows the simpler form
    c0 + C1/n_j + C2/(n_j)^2 + C3/(n_j)^3 + ...
where the coefficients c0, C1, C2, C3, etc. do not depend on the parity of j, and simple numerical methods can be used to evaluate those coefficients. E.g., if we use D=10 (or more), and if each of the values t_j is computed with a little more than 100 digits of precision, then c0 = -0.51541845582541799... can be obtained to 100 digits of precision by applying simple numerical methods to accelerate the convergence of the 21 values t_1, t_2, ..., t_21. (End)

			-0.51541845... (in reference only -0.5154, next digits computed by _Vaclav Kotesovec_).
-0.51541845582541799133011919963942987110456791854801... - _Jon E. Schoenfield_, Jul 08 2015

A. P. Prudnikov, Yu. A. Brychkov and O. I. Marichev, "Integrals and Series", Volume 1: "Elementary Functions", Chapter 5: "Series", New York, Gordon and Breach Science Publishers, 1986-1992, p. 652, formula 6.

Jon E. Schoenfield, Table of n, a(n) for n = 0..100
R. J. Mathar, Approximations to sum_{n>=1} (-1)^floor(n*sqrt 2)/n

Cf. A001951, A006337, A184774.

Sum_{n>=1} (-1)^floor(n*sqrt(2))/n.

cp's OEIS Frontend

A184796 Primes of the form floor(k*sqrt(3)).

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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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A184777 Primes of the form 2k + floor(k*sqrt(2)).

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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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A184802 Primes of the form floor(k*sqrt(5)).

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A184792 Numbers k such that floor(k*r) is prime, where r = golden ratio=(1+sqrt(5))/2.

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