A052035 Palindromic primes whose sum of squared digits is also prime.

11, 101, 131, 191, 313, 353, 373, 797, 919, 10301, 11311, 12721, 13331, 13931, 14341, 14741, 16361, 17971, 18181, 19391, 30103, 30703, 33533, 71317, 71917, 74747, 75557, 76367, 77977, 79397, 90709, 93139, 93739, 95959, 96769, 97379

Offset: 1

Author

Patrick De Geest, Dec 15 1999

Keywords

• nonn
• base

Comments

From Bernard Schott, Oct 20 2021: (Start)

Except for 11, all terms have an odd number of digits.

Except for terms of the form 10^k+1, k >= 2, the middle digit is always odd; the unique known term of the form 10^k+1 for 2 <= k <= 100000 is 101 (see comment in A000533). (End)

Examples

			373 -> 3^2 + 7^2 + 3^2 = 67, which is prime.
		

References

• Charles W. Trigg, Journal of Recreational Mathematics, Vol. 20(2), 1988.

Links

• Michel Marcus, Table of n, a(n) for n = 1..1215
• Mike Mudge, Morph code, Hands On Numbers Count, Personal Computer World, May 1997, p. 290.

Crossrefs

Cf. A000533, A002385, A052034, A003132.

Programs

Mathematica

Select[Prime@ Range[2, 10^4], And[PalindromeQ@ #, PrimeQ@ Total[IntegerDigits[#]^2]] &] (* Michael De Vlieger, Oct 20 2021 *)

PARI

isok(p) = my(d=digits(p)); isprime(p) && (d==Vecrev(d)) && isprime(sum(k=1, #d, d[k]^2)); \\ Michel Marcus, Oct 17 2021

Python

from sympy import isprime
def ok(n):
    s = str(n)
    return s==s[::-1] and isprime(n) and isprime(sum(int(d)**2 for d in s))
print([k for k in range(10**5) if ok(k)]) # Michael S. Branicky, Nov 23 2021

Python

# second version for going to large terms
from sympy import isprime
from itertools import product
def ok(pal):
    return isprime(pal) and isprime(sum(int(d)**2 for d in str(pal)))
def agentod(maxdigs):
    yield 11
    for d in range(3, maxdigs+1, 2):
        pal = 10**(d-1) + 1
        if ok(pal): yield pal
        for first in "1379":
            for left in product("0123456789", repeat=(d-3)//2):
                left = "".join(left)
                for mid in "13579":
                    pal = int(first + left + mid + left[::-1] + first)
                    if ok(pal): yield pal
print([an for an in agentod(5)]) # Michael S. Branicky, Nov 23 2021