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

1,1

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)

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.

Example

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

Mathematica

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

Prog

(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

Crossrefs

Cf. A000533, A002385, A052034, A003132.

Sequence in context: A088281 A068685 A109830 * A083144 A058411 A134462

Adjacent sequences: A052032 A052033 A052034 * A052036 A052037 A052038

Keyword

nonn,base

Author

Patrick De Geest, Dec 15 1999

Status

approved