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

1,1

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.

Links

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

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.

Example

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

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 *)

Prog

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

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.

Sequence in context: A104070 A019371 A068335 * A087776 A098031 A179208

Adjacent sequences: A306763 A306764 A306765 * A306767 A306768 A306769

Keyword

nonn,base

Author

William C. Laursen, Mar 08 2019

Status

approved