A226019 Primes whose binary reversal is a square.

2, 19, 79, 149, 569, 587, 1237, 2129, 2153, 2237, 2459, 2549, 4129, 4591, 4657, 4999, 8369, 8999, 9587, 9629, 9857, 10061, 17401, 17659, 17737, 18691, 20149, 20479, 33161, 33347, 34631, 35117, 35447, 39023, 40427, 40709, 66403, 68539, 74707, 75703, 79063, 79333, 80071

Offset: 1

Author

Alex Ratushnyak, May 23 2013

Keywords

• nonn
• base

Comments

The sequence of corresponding squares begins: 1, 25, 121, 169, 625, 841, 1369, 2209, 2401, 3025, 3481, 2809, 4225, 7921, ...

For n>1 the second and third most significant bits of a(n) are "0" because all odd squares are equal to 1 mod 8. - Andres Cicuttin, May 12 2016

Links

• Chai Wah Wu, Table of n, a(n) for n = 1..6182

Crossrefs

Cf. A007488, A074832.

Subsequence of A204219. Cf. also A235027.

Programs

Mathematica

Select[Table[Prime[j],{j,1,10000}],Element[Sqrt[FromDigits[Reverse[IntegerDigits[#,2]],2]],Integers]&] (* Andres Cicuttin, May 12 2016 *)

PARI

isok(k) = isprime(k) && issquare(fromdigits(Vecrev(binary(k)), 2)); \\ Michel Marcus, Feb 19 2021

Python

import math
primes = []
def addPrime(k):
  for p in primes:
    if k%p==0:  return
    if p*p > k:  break
  primes.append(k)
  r = 0
  p = k
  while k:
    r = r*2 + (k&1)
    k>>=1
  s = int(math.sqrt(r))
  if s*s == r:  print(p, end=', ')
addPrime(2)
addPrime(3)
for i in range(5, 1000000000, 6):
  addPrime(i)
  addPrime(i+2)

Python

from sympy import isprime
A226019_list, i, j = [2], 0, 0
while j < 2**34:
    p = int(format(j,'b')[::-1],2)
    if j % 2 and isprime(p):
        A226019_list.append(p)
    j += 2*i+1
    i += 1
A226019_list = sorted(A226019_list) # Chai Wah Wu, Dec 20 2015

Python

from sympy import integer_nthroot, primerange
def ok(p): return integer_nthroot(int(bin(p)[:1:-1], 2), 2)[1]
def aupto(lim): return [p for p in primerange(2, lim+1) if ok(p)]
print(aupto(80071)) # Michael S. Branicky, Feb 19 2021