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A306766 Primes whose index is divisible by the product of its digits.

%I A306766 #30 Mar 11 2019 09:53:14
%S A306766 11,13,17,61,73,113,223,541,571,1151,1213,1321,1511,1811,2111,2267,
%T A306766 3221,3271,4211,4621,5443,11251,11813,12211,12553,13163,17123,17351,
%U A306766 19211,21143,21713,24137,28181,29921,31511,32213,34141,34361,41141,61129,63211,71263,95231
%N A306766 Primes whose index is divisible by the product of its digits.
%C A306766 It is unknown whether this sequence is finite or not. For instance, if the index is exactly the product of the digits, A097223, it is known that only three such primes exist.
%H A306766 Giovanni Resta, <a href="/A306766/b306766.txt">Table of n, a(n) for n = 1..10000</a> (first 600 terms from Robert Israel)
%F A306766 If a prime is to be in this sequence, its index q must obey A007954(A000040(q))/q = k, where k is an integer.
%e A306766 A000040(21)=73 and 7*3 divides 21.
%e A306766 A000040(30)=113 and 1*1*3 divides 30.
%p A306766 p:= 2: count:= 0: Res:= NULL:
%p A306766 for i from 2 while count < 100 do
%p A306766   p:= nextprime(p);
%p A306766   pd:= convert(convert(p,base,10),`*`);
%p A306766   if pd > 0 and i mod pd = 0 then
%p A306766     count:= count+1; Res:= Res, p
%p A306766   fi
%p A306766 od:
%p A306766 Res; # _Robert Israel_, Mar 10 2019
%t A306766 seqQ[n_] := PrimeQ[n] && (prod=Times@@IntegerDigits[n])>0 && Divisible[PrimePi[n], prod]; Select[Range[100000], seqQ] (* _Amiram Eldar_, Mar 11 2019 *)
%o A306766 (PARI) isok(n) = isprime(n) && (pd=vecprod(digits(n))) && !(primepi(n) % pd); \\ _Michel Marcus_, Mar 09 2019
%Y A306766 A097223 is a subset of this sequence where k=1, k being the above integer found after dividing.
%Y A306766 A004022, the prime repunits, is a subsequence, because the product of the digits for all of them is 1, which trivially divides every index that the prime could hold.
%K A306766 nonn,base
%O A306766 1,1
%A A306766 _William C. Laursen_, Mar 08 2019