A046713 Multiplicative and additive primes: primes where the product and sum of digits are also prime.

2, 3, 5, 7, 113, 131, 151, 311, 2111, 11113, 11117, 11131, 11171, 11311, 111121, 111211, 112111, 1111151, 1111711, 1117111, 1171111, 111111113, 111111131, 111113111, 115111111, 131111111, 1111111121, 1111211111, 1121111111, 11111111113, 11111111131, 11113111111

Offset: 1

Author

Felice Russo

Keywords

• nonn
• base

Comments

Any term of this sequence has one prime digit and all other digits are 1. - Sean A. Irvine, Apr 17 2021

References

• James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, page 113.

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..1000

Crossrefs

Intersection of A028834, A028842, and A000040.

Intersection of A046703 and A046704.

Programs

Mathematica

d[n_]:=IntegerDigits[n]; t={}; Do[p=Prime[n]; If[PrimeQ[Plus@@(x=d[p])]&&PrimeQ[Times@@x],AppendTo[t,p]],{n,2*10^5}]; t (* Jayanta Basu, May 18 2013 *)
Select[Prime[Range[5033*10^5]],AllTrue[{Total[IntegerDigits[ #]],Times@@ IntegerDigits[ #]},PrimeQ]&] (* or -- much faster *) Select[Union[ Flatten[ Table[FromDigits/@Permutations[PadRight[{p},n,1]],{p,{2,3,5,7}},{n,11}]]],AllTrue[{#,Total[ IntegerDigits[#]],Times@@ IntegerDigits[ #]},PrimeQ]&] (* Harvey P. Dale, Feb 28 2022 *)

Extensions

More terms from Harvey P. Dale, Aug 23 2000

Corrected by Jud McCranie, Jan 03 2001

Edited by Charles R Greathouse IV, Aug 02 2010