A065439 Numbers n such that s[n], p[n] and s[n]+p[n] are all prime numbers, where s[n] is the sum of digits of n and p[n] is the product of digits of n.

12, 21, 1112, 1121, 1211, 2111, 1111111112, 1111111121, 1111111211, 1111112111, 1111121111, 1111211111, 1112111111, 1121111111, 1211111111, 2111111111, 1111111111111112, 1111111111111121, 1111111111111211, 1111111111112111

Offset

1,1

Comments

p[n] will always be 2 and s[n] and s[n]+p[n] will always be twin primes.

It is therefore hugely more efficient to search for terms by looking at permutations of any given number of ones together with one two. In addition, it is more efficient not to compute or consider permutations of two together with 4, 7, 10, . . . 3n+1 ones because the sums of those digits are all multiples of 3 so the numbers cannot be primes. - Harvey P. Dale, Apr 27 2019

Links

Harvey P. Dale, Table of n, a(n) for n = 1..1000

A. G. Gevisier, A curiosity about the number 12

Mathematica

Do[s = Apply[ Plus, IntegerDigits[n]]; p = Apply[ Times, IntegerDigits[n]]; If[ PrimeQ[s] && PrimeQ[p] && PrimeQ[s + p], Print[n]], {n, 1, 10^7} ]
spQ[n_]:=Module[{idn=IntegerDigits[n], s, p}, s=Total[idn]; p=Times@@idn; AllTrue[ {s, p, s+p}, PrimeQ]]; Table[Select[FromDigits/@Permutations[ Join[ {2}, PadRight[{}, n, 1]]], spQ], {n, Drop[Range[200], {0, -1, 3}]}]//Flatten//Sort (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Apr 27 2019 *)

Crossrefs

Sequence in context: A292523 A162391 A114015 * A214218 A323084 A323083

Adjacent sequences: A065436 A065437 A065438 * A065440 A065441 A065442

Keyword

base,nonn

Author

Santi Spadaro, Nov 17 2001

Extensions

Comment and more terms from Larry Reeves (larryr(AT)acm.org), Nov 27 2001

Status

approved