A095290 -id:A095290 - cp's OEIS frontend

A095280 Lower Wythoff primes, i.e., primes in A000201.

3, 11, 17, 19, 29, 37, 43, 53, 59, 61, 67, 71, 79, 97, 101, 103, 113, 127, 131, 137, 139, 163, 173, 179, 181, 197, 199, 211, 223, 229, 239, 241, 257, 263, 271, 281, 283, 307, 313, 317, 331, 347, 349, 359, 367, 373, 383, 389, 401, 409, 419, 433

Offset: 1

Antti Karttunen, Jun 04 2004

nonn

Contains all primes p whose Zeckendorf-expansion A014417(p) ends with an even number of 0's.

For generalizations and conjectures, see A184774.

Robert Israel, Table of n, a(n) for n = 1..10000
Antti Karttunen and J. Moyer, C-program for computing the initial terms of this sequence

Intersection of A000040 & A000201. Complement of A095281 in A000040. Cf. A095080, A095083, A095084, A095290, A184792, A184793, A184794, A184796.

R:= NULL: count:= 0:
for n from 1 while count < 100 do
  p:= floor(n*phi);
  if isprime(p) then R:= R,p; count:= count+1 fi
od:
R; # Robert Israel, Jan 17 2023

Mathematica
```
(See A184792.)
```

from math import isqrt
from itertools import count, islice
from sympy import isprime
def A095280_gen(): # generator of terms
    return filter(isprime,((n+isqrt(5*n**2)>>1) for n in count(1)))
A095280_list = list(islice(A095280_gen(),30)) # Chai Wah Wu, Aug 16 2022

A095281 Upper Wythoff primes, i.e., primes in A001950.

Original entry on oeis.org

2, 5, 7, 13, 23, 31, 41, 47, 73, 83, 89, 107, 109, 149, 151, 157, 167, 191, 193, 227, 233, 251, 269, 277, 293, 311, 337, 353, 379, 397, 421, 431, 439, 463, 479, 523, 541, 547, 557, 599, 607, 617, 641, 659, 683, 691, 701, 709, 719, 727, 733, 743

Offset: 1

Antti Karttunen, Jun 04 2004

nonn

Contains all primes p whose Zeckendorf-expansion A014417(p) ends with an odd number of 0's.

Antti Karttunen and J. Moyer, C-program for computing the initial terms of this sequence

Intersection of A000040 & A001950. Complement of A095280 in A000040. Cf. A095081, A095083, A095084, A095290.

from math import isqrt
from itertools import count, islice
from sympy import isprime
def A095281_gen(): # generator of terms
    return filter(isprime,((n+isqrt(5*n**2)>>1)+n for n in count(1)))
A095281_list = list(islice(A095281_gen(),30)) # Chai Wah Wu, Aug 16 2022

A095291 Number of upper Wythoff primes (A095281) in range ]2^n,2^(n+1)].

Original entry on oeis.org

0, 2, 1, 2, 2, 5, 9, 13, 35, 51, 93, 175, 331, 595, 1149, 2182, 4097, 7749, 14780, 28131, 53583, 102372, 196345, 375876, 722743, 1392156, 2684022, 5180823, 10008419, 19368226, 37499404, 72698062, 141064116

Offset: 1

Antti Karttunen, Jun 04 2004

nonn

As expected, the ratio of a(n)/A036378(n) seems to approach 1-((sqrt(5)-1)/2) (= 0.381966011250...): 0, 1, 0.5, 0.4, 0.285714, 0.384615, 0.391304, 0.302326, 0.466667, 0.372263, 0.364706, 0.377155, 0.379587, 0.369107, 0.379208, 0.382204, 0.381152, 0.380039, 0.382555, 0.382287, 0.381819, 0.381677, 0.382211, 0.381283, 0.381572, 0.381858, 0.381943, 0.382013, 0.381895, 0.382035, 0.381935, 0.381947, 0.381953

Also expected, the ratio a(n)/A095061(n) seems to approach 1: 1, 0, 0, 1, 0.5, 1.25, 1.28571, 0.72222, 1.4, 0.94444, 0.88571, 0.98315, 0.99699, 0.93407, 0.98627, 0.99453, 0.98462, 0.9973, 0.99865, 1.0011, 1.00108, 0.99979, 1.00208, 0.99622, 0.99835, 1.00039, 0.99973, 1.00046, 0.99983, 1.00031, 0.99994, 0.99994, 1.00001

A. Karttunen and J. Moyer: C-program for computing the initial terms of this sequence
Index entries for sequences related to occurrences of various subsets of primes in range ]2^n,2^(n+1)]

a(n) = A036378(n)-A095290(n). Cf. A095061, A095290.

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A095280 Lower Wythoff primes, i.e., primes in A000201.

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A095281 Upper Wythoff primes, i.e., primes in A001950.

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Python

A095291 Number of upper Wythoff primes (A095281) in range ]2^n,2^(n+1)].

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