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A364479 Happy palindromic primes.

%I A364479 #52 Feb 16 2025 08:34:06
%S A364479 7,313,383,11311,15451,30103,30803,35053,36263,71317,74047,94349,
%T A364479 94649,95959,98689,1221221,1257521,1262621,1281821,1311131,1444441,
%U A364479 1551551,1594951,1597951,1658561,1703071,1737371,1764671,1829281,1924291,1957591,1970791,1981891,1988891,3001003
%N A364479 Happy palindromic primes.
%C A364479 All terms in the sequence are prime numbers that read the same backward and forward. Each term is a happy number, meaning it converges to 1 under the process of repeatedly summing the squares of its digits. The sequence is a subset of both the palindromic prime numbers (A002385) and the happy numbers (A007770).
%C A364479 There are no happy palindromic primes with an even number of digits: every palindromic number with an even number of digits is divisible by 11, so 11 itself is the only palindromic prime, and it is not a happy number.
%H A364479 Chai Wah Wu, <a href="/A364479/b364479.txt">Table of n, a(n) for n = 1..10000</a>
%H A364479 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/HappyNumber.html">Happy Number</a>.
%H A364479 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PalindromicPrime.html">Palindromic Prime</a>.
%e A364479 313 is a term as it is palindromic (can be reversed), is a prime and is happy: 3^2 + 1^2 + 3^2 = 19, 1^2 + 9^2 = 82, 8^2 + 2^2 = 68, 6^2 + 8^2 = 100, 1^2 + 0^2 + 0^2 = 1.
%t A364479 happyQ[n_] := NestWhile[Plus @@ (IntegerDigits[#]^2) &, n, UnsameQ, All] == 1; Select[Prime[Range[220000]], PalindromeQ[#] && happyQ[#] &] (* _Amiram Eldar_, Jul 28 2023 *)
%o A364479 (Python)
%o A364479 def is_prime(num):
%o A364479     return num > 1 and all(num % i != 0 for i in range(2, int(num ** 0.5) + 1))
%o A364479 def is_palindrome(num):
%o A364479     return str(num) == str(num)[::-1]
%o A364479 def is_happy(num):
%o A364479     while num != 1 and num != 4:
%o A364479         num = sum(int(digit) ** 2 for digit in str(num))
%o A364479     return num == 1
%o A364479 happy_palindromic_primes = [num for num in range(1, 10000000) if is_prime(num) and is_palindrome(num) and is_happy(num)]
%o A364479 print(happy_palindromic_primes)
%o A364479 (Python)
%o A364479 from itertools import islice
%o A364479 from sympy import isprime
%o A364479 def A364479_gen(): # generator of terms
%o A364479     n = 1
%o A364479     while True:
%o A364479         for z in (1,3,5,7,9):
%o A364479             for y in range(z*n, (z+1)*n):
%o A364479                 k, m = y//10, 0
%o A364479                 while k >= 10:
%o A364479                     k, r = divmod(k, 10)
%o A364479                     m = 10*m + r
%o A364479                 if isprime(a:=y*n + 10*m + k):
%o A364479                     b = a
%o A364479                     while b not in {1,37,58,89,145,42,20,4,16}:
%o A364479                         b = sum((0, 1, 4, 9, 16, 25, 36, 49, 64, 81)[ord(d)-48] for d in str(b))
%o A364479                     if b == 1:
%o A364479                         yield a
%o A364479         n *= 10
%o A364479 A364479_list = list(islice(A364479_gen(),20)) # _Chai Wah Wu_, Aug 02 2023
%Y A364479 Intersection of A002385 and A007770.
%K A364479 nonn,base
%O A364479 1,1
%A A364479 _Simon R Blow_, Jul 26 2023