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A267277 Zeroless primes p such that p*(product of digits of p)+(sum of digits of p) is also prime.

%I A267277 #21 Dec 20 2018 18:24:28
%S A267277 11,13,17,19,31,37,43,47,61,73,79,83,223,227,263,281,283,463,643,683,
%T A267277 821,827,881,1117,1231,1259,1291,1321,1361,1367,1433,1471,1543,1567,
%U A267277 1583,1597,1619,1637,1657,1699,1723,1741,1753,1777,1933,1951,1973
%N A267277 Zeroless primes p such that p*(product of digits of p)+(sum of digits of p) is also prime.
%C A267277 Zeroless means that the decimal expansion has no digit "0", so no element of A056709 is in the sequence.
%C A267277 If we define a function "n*times products of digits plus sum of digits", f(n) = n*A007954(n) + A007953(n), then iterating the function starting at 217421 generates a chain of at least 4 primes: 217421 -> 24351169 -> 157795575151 -> 1522234189034803183.
%H A267277 Robert Israel, <a href="/A267277/b267277.txt">Table of n, a(n) for n = 1..10000</a>
%e A267277 19 => 19*1*9+1+9 = 181 (is prime).
%e A267277 821 => 821*8*2*1+8+2+1 = 13147 (is prime).
%e A267277 2357 => 2357*2*3*5*7+2+3+5+7 = 494987 (is prime).
%e A267277 99995999 => 99995999*(9^7)*5+9*7+5 = 2391388816705223 (is prime).
%p A267277 isA267277 := proc(n)
%p A267277     local pdgs ;
%p A267277     if isprime(n) then
%p A267277         pdgs := A007954(n) ;
%p A267277         if pdgs <> 0 then
%p A267277             isprime(n*pdgs+A007953(n)) ;
%p A267277         else
%p A267277             false;
%p A267277         end if;
%p A267277     else
%p A267277         false;
%p A267277     end if;
%p A267277 end proc:
%p A267277 for n from 1 to 400 do
%p A267277     if isA267277(n) then
%p A267277         printf("%d,\n",n);
%p A267277     end if;
%p A267277 end do: # _R. J. Mathar_, Jan 16 2016
%t A267277 Select[Prime@ Range@ 480, And[Last@ DigitCount@ # == 0, PrimeQ[Function[k, # Times @@ k + Total@ k]@ IntegerDigits@ #]] &] (* _Michael De Vlieger_, Jan 12 2016 *)
%Y A267277 Cf. A007953, A007954, A038618, A056709.
%K A267277 nonn,base,less,easy
%O A267277 1,1
%A A267277 _Emre APARI_, Jan 12 2016