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A306766
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Primes whose index is divisible by the product of its digits.
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2
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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
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OFFSET
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1,1
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COMMENTS
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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.
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LINKS
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FORMULA
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If a prime is to be in this sequence, its index q must obey A007954(A000040(q))/q = k, where k is an integer.
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EXAMPLE
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A000040(30)=113 and 1*1*3 divides 30.
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MAPLE
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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:
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MATHEMATICA
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seqQ[n_] := PrimeQ[n] && (prod=Times@@IntegerDigits[n])>0 && Divisible[PrimePi[n], prod]; Select[Range[100000], seqQ] (* Amiram Eldar, Mar 11 2019 *)
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PROG
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(PARI) isok(n) = isprime(n) && (pd=vecprod(digits(n))) && !(primepi(n) % pd); \\ Michel Marcus, Mar 09 2019
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CROSSREFS
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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.
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KEYWORD
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nonn,base
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AUTHOR
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STATUS
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approved
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