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

%I A052035 #39 Nov 23 2021 15:58:54
%S A052035 11,101,131,191,313,353,373,797,919,10301,11311,12721,13331,13931,
%T A052035 14341,14741,16361,17971,18181,19391,30103,30703,33533,71317,71917,
%U A052035 74747,75557,76367,77977,79397,90709,93139,93739,95959,96769,97379
%N A052035 Palindromic primes whose sum of squared digits is also prime.
%C A052035 From _Bernard Schott_, Oct 20 2021: (Start)
%C A052035 Except for 11, all terms have an odd number of digits.
%C A052035 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)
%D A052035 Charles W. Trigg, Journal of Recreational Mathematics, Vol. 20(2), 1988.
%H A052035 Michel Marcus, <a href="/A052035/b052035.txt">Table of n, a(n) for n = 1..1215</a>
%H A052035 Mike Mudge, <a href="https://archive.org/details/PersonalComputerWorldMagazine/PCW%20199705%20May%20Created%20From%20PCW%20Cover%20CD/page/n121/mode/1up?view=theater">Morph code</a>, Hands On Numbers Count, Personal Computer World, May 1997, p. 290.
%e A052035 373 -> 3^2 + 7^2 + 3^2 = 67, which is prime.
%t A052035 Select[Prime@ Range[2, 10^4], And[PalindromeQ@ #, PrimeQ@ Total[IntegerDigits[#]^2]] &] (* _Michael De Vlieger_, Oct 20 2021 *)
%o A052035 (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
%o A052035 (Python)
%o A052035 from sympy import isprime
%o A052035 def ok(n):
%o A052035     s = str(n)
%o A052035     return s==s[::-1] and isprime(n) and isprime(sum(int(d)**2 for d in s))
%o A052035 print([k for k in range(10**5) if ok(k)]) # _Michael S. Branicky_, Nov 23 2021
%o A052035 (Python) # second version for going to large terms
%o A052035 from sympy import isprime
%o A052035 from itertools import product
%o A052035 def ok(pal):
%o A052035     return isprime(pal) and isprime(sum(int(d)**2 for d in str(pal)))
%o A052035 def agentod(maxdigs):
%o A052035     yield 11
%o A052035     for d in range(3, maxdigs+1, 2):
%o A052035         pal = 10**(d-1) + 1
%o A052035         if ok(pal): yield pal
%o A052035         for first in "1379":
%o A052035             for left in product("0123456789", repeat=(d-3)//2):
%o A052035                 left = "".join(left)
%o A052035                 for mid in "13579":
%o A052035                     pal = int(first + left + mid + left[::-1] + first)
%o A052035                     if ok(pal): yield pal
%o A052035 print([an for an in agentod(5)]) # _Michael S. Branicky_, Nov 23 2021
%Y A052035 Cf. A000533, A002385, A052034, A003132.
%K A052035 nonn,base
%O A052035 1,1
%A A052035 _Patrick De Geest_, Dec 15 1999