A107650 Numbers n such that both numbers n/(d_1*d_2* ...*d_k) and n/(d_1+d_2+ ... +d_k) are prime, where d_1 d_2 ... d_k is the decimal expansion of n.

11133, 11331, 13131, 31113, 112116, 121116, 13111212, 111311115, 11114112112, 111212112112, 1111111711311, 1111171111113, 11111111112611112, 11111111121161112, 11111112111161112, 11111119111131111, 11111131111119111, 11111139111111111, 11111193111111111, 11111211161111112, 11111611111211112, 11116111112111112, 11116111211111112

Offset: 1

Author

Farideh Firoozbakht, May 21 2005

Keywords

• base
• nonn

Comments

For n in this sequence, let prime p = n/(d_1*d_2* ...*d_k) so that n = d_1*d_2* ...*d_k * p. Then n/(d_1+d_2+ ... +d_k) equals either p or some prime dividing d_1*d_2* ...*d_k, that is 2, 3, 5, or 7. The latter case never takes place and thus n/(d_1*d_2* ...*d_k) = n/(d_1+d_2+ ... +d_k) is the same prime. So this sequence is a subsequence of both A034710 and A066307. - Max Alekseyev, Aug 19 2013

Examples

			111311115 is in the sequence because
111311115/(1*1*1*3*1*1*1*1*5) and 111311115/(1+1+1+3+1+1+1+1+5)
are prime(since 1*1*1*3*1*1*1*1*5=1+1+1+3+1+1+1+1+5, the primes are equal).
		

Links

• Chai Wah Wu, Table of n, a(n) for n = 1..10000 (terms 1..1717 from Max Alekseyev).

Programs

Mathematica

Do[h = IntegerDigits[m]; l = Length[h]; If[Min[h] > 0 && PrimeQ[m/Sum[h[[k]], {k, l}]] && PrimeQ[m/Product[ h[[k]], {k, l}]], Print[m]], {m, 265000000}]

Extensions

a(9)-a(10) from Sean A. Irvine, Nov 28 2010

Terms a(11) onward from Max Alekseyev, Aug 20 2013