A007488 -id:A007488 - cp's OEIS frontend

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

Alex Ratushnyak, May 23 2013

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

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

Cf. A007488, A074832.

Subsequence of A204219. Cf. also A235027.

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

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

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)

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

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

A362677 Primes whose reversal + 1 is a cube.

Original entry on oeis.org

7, 421, 827, 4733, 32831, 57571, 228301, 364751, 892079, 1932677, 2256713, 3684211, 4213591, 6751853, 7218259, 7887707, 8497033, 15720487, 19925251, 21055813, 28756943, 29547961, 47369149, 51881849, 55033973, 57954643, 59677001, 63062963, 74415157, 88535987

Offset: 1

Zhining Yang, Jul 03 2023

From Jon E. Schoenfield, Jul 03 2023: (Start)
Equivalently, primes whose reversal is one less than the cube of a positive integer whose last digit is not a 1.
Since no prime starts or ends with a 0, reversing the prime will not change the number of digits, and since no prime consists only of 9's, adding 1 to its reversal will not change the number of digits, either, so the cube will have the same number of digits as the prime. Since the prime cannot begin with a 0, its reversal cannot end in 0, so the cube cannot end in 1 (and a cube ends in 1 if and only if its cube root ends in 1). Since cubes are less dense than primes, a reasonably efficient but simple way to search for all terms having at most D digits is to test each positive integer r < 10^(D/3) such that r mod 10 != 1: if the reversal of r^3 - 1 is a prime, then that prime is a term of the sequence. (End)

			421 is prime and reversal(421) + 1 = 124 + 1 = 125 = 5^3.
364751 is in the sequence because it is prime and reversal(364751) + 1 = 157463 + 1 = 157464 = 54^3.

Michael S. Branicky, Table of n, a(n) for n = 1..10000

Cf. A007488, A167217.

Select[Prime@Range@1000000,IntegerQ@CubeRoot@(FromDigits@Reverse@IntegerDigits@#+1) &]
r = Select[Range@300, Mod[#, 3] != 1 && Mod[#, 10] != 1 &];
s = Sort@Select[FromDigits /@ Reverse /@ IntegerDigits@(r^3 - 1),PrimeQ]
Select[Prime[Range[514*10^4]],IntegerQ[CubeRoot[1+IntegerReverse[#]]]&] (* Harvey P. Dale, Apr 03 2025 *)

from sympy import isprime
s=[int(str(k**3-1)[::-1]) for k in range(1,301)if k%10!=1 and k%3!=1]
t=[p for p in s if isprime(p)]
t.sort
print(t)

a(18)-a(30) from Jon E. Schoenfield, Jul 03 2023

A087768 Minimal prime whose reversal is a multiple of n, or 0 if no such prime exists.

Original entry on oeis.org

2, 2, 3, 23, 5, 0, 7, 23, 0, 0, 11, 0, 19, 41, 0, 23, 43, 0, 59, 0, 0, 0, 29, 0, 521, 269, 0, 211, 457, 0, 13, 23, 0, 43, 53, 0, 47, 67, 0, 0, 349, 0, 103, 0, 0, 29, 257, 0, 89, 0, 0, 401, 109, 0, 0, 211, 0, 457, 449, 0, 397, 421, 0, 821, 523, 0, 431, 631, 0, 0, 17, 0, 37, 47, 0

Offset: 1

Zak Seidov, Oct 03 2003

There are no primes whose reversal is a multiple of 3, 10, 11 (except 3 & 11).

Cf. A007488.

Corrected and extended by Ray Chandler, Oct 05 2003

A087769 Minimal prime whose reversal is a multiple of n-th prime.

Original entry on oeis.org

2, 3, 5, 7, 11, 19, 43, 59, 29, 457, 13, 47, 349, 103, 257, 109, 449, 397, 431, 17, 37, 97, 233, 653, 79, 101, 127, 701, 367, 197, 3863, 131, 1627, 379, 547, 151, 587, 1667, 433, 643, 617, 181, 191, 277, 887, 599, 3709, 1973, 809, 619, 239, 659, 1747, 1877, 1433

Offset: 1

Zak Seidov, Oct 03 2003

			n=100, p(n)=541, minimal prime whose reversal is a multiple of 541 is 2801.

Cf. A007488.

Module[{nn=600,irp},irp=IntegerReverse/@Prime[Range[nn]];Table[ IntegerReverse[ SelectFirst[irp,IntegerQ[#/p]&]],{p,Prime[Range[60]]}]] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Sep 26 2020 *)

A345956 Primes whose reverse is the square of a Fibonacci number.

Original entry on oeis.org

1297, 921241, 186331288599111530713519, 98866853184420632874393761586063024530467395043

Offset: 1

Tanya Khovanova, Jun 30 2021

			The reverse of 1297 is 7921 = 89^2, where 89 is a Fibonacci number. Thus 1297 is in this sequence.

Cf. A007488 (primes whose reversal is a square), A036797 (Iccanobif primes).

Cf. A000045, A007598.

Select[IntegerReverse/@(Fibonacci@Range@1000^2),PrimeQ] (* Giorgos Kalogeropoulos, Jun 30 2021 *)

for(k=3,200,my(f=fibonacci(k)^2,fi=fromdigits(Vecrev(digits(f))));if(isprime(fi),print1(fi,", "))) \\ Hugo Pfoertner, Jun 30 2021

from sympy import fibonacci, isprime
def rev(n): return int(str(n)[::-1])
def auptok(klimit): return [t for t in (rev(fibonacci(k)**2) for k in range(3, klimit+1)) if isprime(t)]
print(auptok(200)) # Michael S. Branicky, Jun 30 2021

cp's OEIS Frontend

A226019 Primes whose binary reversal is a square.

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A362677 Primes whose reversal + 1 is a cube.

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A087768 Minimal prime whose reversal is a multiple of n, or 0 if no such prime exists.

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A087769 Minimal prime whose reversal is a multiple of n-th prime.

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A345956 Primes whose reverse is the square of a Fibonacci number.

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Mathematica

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Python