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

11, 13, 17, 19, 31, 37, 43, 47, 61, 73, 79, 83, 223, 227, 263, 281, 283, 463, 643, 683, 821, 827, 881, 1117, 1231, 1259, 1291, 1321, 1361, 1367, 1433, 1471, 1543, 1567, 1583, 1597, 1619, 1637, 1657, 1699, 1723, 1741, 1753, 1777, 1933, 1951, 1973

Offset: 1

Author

Emre APARI, Jan 12 2016

Keywords

• nonn
• base
• less
• easy

Comments

Zeroless means that the decimal expansion has no digit "0", so no element of A056709 is in the sequence.

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.

Examples

			19 => 19*1*9+1+9 = 181 (is prime).
821 => 821*8*2*1+8+2+1 = 13147 (is prime).
2357 => 2357*2*3*5*7+2+3+5+7 = 494987 (is prime).
99995999 => 99995999*(9^7)*5+9*7+5 = 2391388816705223 (is prime).
		

Links

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

Crossrefs

Cf. A007953, A007954, A038618, A056709.

Programs

Maple

isA267277 := proc(n)
    local pdgs ;
    if isprime(n) then
        pdgs := A007954(n) ;
        if pdgs <> 0 then
            isprime(n*pdgs+A007953(n)) ;
        else
            false;
        end if;
    else
        false;
    end if;
end proc:
for n from 1 to 400 do
    if isA267277(n) then
        printf("%d,\n",n);
    end if;
end do: # R. J. Mathar, Jan 16 2016

Mathematica

Select[Prime@ Range@ 480, And[Last@ DigitCount@ # == 0, PrimeQ[Function[k, # Times @@ k + Total@ k]@ IntegerDigits@ #]] &] (* Michael De Vlieger, Jan 12 2016 *)