login
A269930
Primes whose digits are all prime, sum of digits is prime and sum of reciprocals of digits is also prime.
0
32233, 32323, 33223, 2222333, 2223233, 2232323, 2233223, 3223223, 272777777, 277727777, 722777777, 772277777, 772777727, 777727277, 777777227, 33333555553, 33355535533, 33355553353, 33533555353, 33553353553, 33553553353, 33553553533, 33555353353, 33555533533, 35335355353, 35335533553, 35353335553
OFFSET
1,1
COMMENTS
Intersection of A019546, A046704, A266815.
Furthermore, 32233, 32323 and 2223233 are primes with prime subscripts (A006450). In fact, 32233 is the 3457th prime, 32323 is the 3467th prime, and 2223233 is the 164239th prime.
EXAMPLE
32233 is prime, its digits are primes (2 and 3), their sum is prime (3 + 2 + 2 + 3 + 3 = 13) and the sum of reciprocal of digits is also prime (1/3 + 1/2 + 1/2 + 1/3 + 1/3 = 2).
MAPLE
P:=proc(q) local a, b, k, ok, ok2, n;
for n from 1 to q do if isprime(n) then ok:=1; a:=0; for k from 0 to ilog10(n) do
if trunc(n/10^k) mod 10>0 then a:=a+1/(trunc(n/10^k) mod 10) else ok:=0; break; fi; od;
if ok=1 and type(a, integer) then if isprime(a) then a:=0; b:=n; ok2:=1;
for k from 1 to ilog10(n)+1 do if isprime(b mod 10) then a:=a+(b mod 10); b:=trunc(b/10);
else ok2:=0; break; fi; od; if ok2=1 and isprime(a) then print(n); fi; fi; fi; fi; od; end: P(10^9);
MATHEMATICA
Select[Select[Flatten@ Map[Map[FromDigits, Tuples[{2, 3, 5, 7}, #]] &, Range@ 11], PrimeQ], And[PrimeQ[Total@ #], PrimeQ[Total[1/#]]] &@ IntegerDigits@ # &] (* Michael De Vlieger, Mar 08 2016 *)
CROSSREFS
KEYWORD
nonn,easy,base
AUTHOR
Paolo P. Lava, Mar 08 2016
EXTENSIONS
More terms from Michael De Vlieger, Mar 08 2016
STATUS
approved