A207293 Primes p whose digit sum s(p) is also prime but whose iterated digit sum s(s(p)) is not prime.

67, 89, 139, 157, 179, 193, 197, 199, 229, 269, 283, 337, 359, 373, 379, 397, 409, 449, 463, 467, 487, 557, 571, 577, 593, 607, 643, 647, 661, 683, 719, 733, 739, 751, 757, 773, 809, 823, 827, 829, 863, 881, 883, 919, 937, 953, 971, 991, 1039, 1093, 1097, 1129, 1187

Offset: 1

Author

Jonathan Sondow, Jun 09 2012

Keywords

• base
• nonn

Comments

A046704 is primes p with s(p) also prime. A207294 is primes p with s(p) and s(s(p)) also prime. A070027 is primes p with all s(p), s(s(p)), s(s(s(p))), ... also prime. A104213 is primes p with s(p) not prime. A213354 is primes p with s(p) and s(s(p)) also prime but s(s(s(p))) not prime. A213355 is smallest prime p whose k-fold digit sum s(s(..s(p)..)) is also prime for all k < n, but not for k = n.

Examples

			67 is prime and s(67) = 6+7 = 13 is also prime, but s(s(67)) = s(13) = 1+3 = 4 is not prime. Since no smaller prime has this property, a(1) = 67.
		

Links

• Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A046704, A070027, A104213, A207294, A213354, A213355.

Programs

Maple

isA207293 := proc(n)
    local d;
    if isprime(n) then
        d := digsum(n) ;
        if isprime(d) then
            d := digsum(d) ;
            if isprime(d) then
                false ;
            else
                true ;
            end if;
        else
            false ;
        end if;
    else
        false;
    end if;
end proc:
A207293 := proc(n)
    option remember ;
    if n = 1 then
        67 ;
    else
        a := nextprime(procname(n-1)) ;
        while not isA207293(a) do
            a := nextprime(a) ;
        end do:
        a ;
    end if;
end proc: # R. J. Mathar, Feb 04 2021

Mathematica

Select[Prime[Range[300]],
PrimeQ[Apply[Plus, IntegerDigits[#]]] && !
    PrimeQ[Apply[Plus, IntegerDigits[Apply[Plus, IntegerDigits[#]]]]] &]
idsQ[n_]:=PrimeQ[Rest[NestList[Total[IntegerDigits[#]]&,n,2]]]=={True,False}; Select[Prime[Range[200]],idsQ] (* Harvey P. Dale, Dec 28 2013 *)

PARI

select(p->my(s=sumdigits(p));isprime(s)&&!isprime(sumdigits(s)), primes(1000)) \\ Charles R Greathouse IV, Jun 10 2012