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

1,1

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

Links

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

Example

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).

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}]

Crossrefs

Sequence in context: A261778 A268278 A073038 * A291705 A244660 A205751

Adjacent sequences: A107647 A107648 A107649 * A107651 A107652 A107653

Keyword

base,nonn

Author

Farideh Firoozbakht, May 21 2005

Extensions

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

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

Status

approved