A353703 Palindromes (A002113) in A157037.

6, 22, 66, 202, 222, 282, 434, 454, 474, 494, 555, 595, 838, 858, 969, 1001, 1551, 1771, 3333, 3553, 5335, 6006, 6226, 6886, 8778, 9889, 12921, 14541, 15051, 16261, 16761, 17171, 18681, 19491, 20202, 20602, 20802, 20902, 24142, 24242, 24542, 28282, 28482, 30003

Offset: 1

Author

Marius A. Burtea, May 08 2022

Keywords

• nonn
• base

Comments

Intersection of A002113 and A157037.

Examples

			22 = A002113(12) and 22 = A157037(3), so 22 is a term.
66 = A002113(16) and 22 = A157037(8), so 66 is a term.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A002113, A003415, A157037.

Programs

Magma

f:=func; pal:=func; [n:n in [2..30003]| pal(n) and IsPrime(Floor(f(n)))];

Maple

filter:= proc(n) local t;
  isprime(n*add(t[2]/t[1], t=ifactors(n)[2]))
end proc:
digrev:= proc(n) local L,i;
  L:= convert(n,base,10);
  add(L[-i]*10^(i-1),i=1..nops(L))
end proc:
N:= 100: # for a(1) to a(N)
Res:= 6: count:= 1:
for d from 2 while count < N do
  if d::even then
    m:= d/2;
    for n from 10^(m-1) to 10^m-1 while count < N do
      v:=  n*10^m + digrev(n);
      if filter(v) then Res:= Res,v; count:= count+1  fi;
    od
  else
    m:= (d-1)/2;
    for n from 10^(m-1) to 10^m-1 while count < N do
      for y from 0 to 9 while count < N do
        v:= n*10^(m+1)+y*10^m+digrev(n);
        if filter(v) then Res:= Res,v; count:= count+1 fi;
    od od:
  fi
od:
Res; # Robert Israel, May 09 2023

Mathematica

d[0] = d[1] = 0; d[n_] := n * Plus @@ ((Last[#]/First[#]) & /@ FactorInteger[n]); Select[Range[30003], PalindromeQ[#] && PrimeQ[d[#]] &] (* Amiram Eldar, May 09 2022 *)

PARI

ad(n) = vecsum([n/f[1]*f[2]|f<-factor(n+!n)~]); \\ A003415
isok(m) = my(d); isprime(ad(m)) && (d=digits(m)) && (d==Vecrev(d)); \\ Michel Marcus, May 09 2022

Python

from itertools import chain, count, islice
from sympy import isprime, factorint
def A353703_gen(): # generator of terms
    return filter(lambda n:isprime(sum(n*e//p for p,e in factorint(n).items())), chain.from_iterable(chain((int((s:=str(d))+s[-2::-1]) for d in range(10**l,10**(l+1))), (int((s:=str(d))+s[::-1]) for d in range(10**l,10**(l+1)))) for l in count(0)))
A353703_list = list(islice(A353703_gen(),20)) # Chai Wah Wu, Jun 23 2022