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

11, 13, 17, 61, 73, 113, 223, 541, 571, 1151, 1213, 1321, 1511, 1811, 2111, 2267, 3221, 3271, 4211, 4621, 5443, 11251, 11813, 12211, 12553, 13163, 17123, 17351, 19211, 21143, 21713, 24137, 28181, 29921, 31511, 32213, 34141, 34361, 41141, 61129, 63211, 71263, 95231

Offset: 1

Author

William C. Laursen, Mar 08 2019

Keywords

• nonn
• base

Comments

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.

Examples

			A000040(21)=73 and 7*3 divides 21.
A000040(30)=113 and 1*1*3 divides 30.
		

Links

• Giovanni Resta, Table of n, a(n) for n = 1..10000 (first 600 terms from Robert Israel)

Crossrefs

A097223 is a subset of this sequence where k=1, k being the above integer found after dividing.

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.

Programs

Maple

p:= 2: count:= 0: Res:= NULL:
for i from 2 while count < 100 do
  p:= nextprime(p);
  pd:= convert(convert(p,base,10),`*`);
  if pd > 0 and i mod pd = 0 then
    count:= count+1; Res:= Res, p
  fi
od:
Res; # Robert Israel, Mar 10 2019

Mathematica

seqQ[n_] := PrimeQ[n] && (prod=Times@@IntegerDigits[n])>0 && Divisible[PrimePi[n], prod]; Select[Range[100000], seqQ] (* Amiram Eldar, Mar 11 2019 *)

PARI

isok(n) = isprime(n) && (pd=vecprod(digits(n))) && !(primepi(n) % pd); \\ Michel Marcus, Mar 09 2019

Formula

If a prime is to be in this sequence, its index q must obey A007954(A000040(q))/q = k, where k is an integer.