A086766 -id:A086766 - cp's OEIS frontend

A086920 Smallest k such that n times concatenation of k with itself followed by a 1 is a prime, or 0 if no such number exists.

1, 3, 2, 3, 3, 3, 3, 30, 6, 4, 5, 5, 25, 42, 17, 19, 2, 1, 7, 6, 6, 1, 50, 29, 138, 153, 4, 45, 54, 7, 15, 9, 78, 147, 95, 58, 102, 18, 3, 115, 54, 6, 46, 9, 24, 64, 183, 120, 3, 33, 112, 46, 105, 4, 279, 122, 68, 75, 3, 22, 294, 56, 92, 76, 6, 19, 94, 71, 12, 75, 30, 16, 78, 320, 33

Offset: 1

Amarnath Murthy, Sep 18 2003

Conjecture: no term is zero.

			a(3) = 2 as 2221 is a prime. a(4) = 3 as 33331 is a prime but 11111 and 22221 are not.
a(8) = 30 and the prime arising is 30303030303030301.

Cf. A087604, A087605, A087606, A087607, A087608, A087609.
Cf. A086923.
Cf. A086766.

More terms from Ray G. Opao, Apr 15 2004

More terms from David Wasserman, Apr 12 2005

A252491 a(n) = (10^(n^2) - 1)/(10^n - 1).

Original entry on oeis.org

1, 101, 1001001, 1000100010001, 100001000010000100001, 1000001000001000001000001000001, 1000000100000010000001000000100000010000001, 100000001000000010000000100000001000000010000000100000001, 1000000001000000001000000001000000001000000001000000001000000001

Offset: 1

M. F. Hasler, Jan 08 2015

When written in base 10, the terms consist of n digits '1' separated by strings of n-1 digits '0'.
This sequence is relevant for counterexamples to a conjecture in A086766: If p is prime and a(p) is not prime, then A086766(10^(p-1)) = 0.
a(n) is the product of A019328(d) for all d that divide n^2 but not n. - Robert Israel, Jan 08 2015
If a(n) is a prime then n is a prime. What is the smallest prime term greater than 101 in this sequence? - Farideh Firoozbakht, Jan 08 2015
According to what precedes, a(n) is prime iff A019328(d) is prime, where d is the only divisor of n^2 which is not a divisor of n, i.e., iff n is a prime and n^2 is in A138940. No such term is known, except for n=2. - M. F. Hasler, Jan 09 2015

Cf. A128889 (for 2 instead of 10).

seq((10^(n^2)-1)/(10^n-1), n=1..20); # Robert Israel, Jan 08 2015

PARI
```
A252491(n)=(10^(n^2)-1)\(10^n-1)
```

A087403 a(n) = smallest prime of the form 10*K(n) + 1, where K is a number obtained by concatenation of n with itself, or 0 if no such prime exists.

Original entry on oeis.org

11, 2221, 31, 41, 555555555551, 61, 71, 881, 991, 101, 1111111111111111111, 1212121, 131, 14141414141, 151, 1616161, 1717171717171717171717171717171, 181, 191, 20201, 211

Offset: 1

Amarnath Murthy, Sep 10 2003

Conjecture: No term is zero.
Next term a(22) is too large (121 digits) to include in sequence. - Ray Chandler, Sep 23 2003
From Farideh Firoozbakht, Jan 07 2015: (Start)
The conjecture is not true. There exist many numbers n such that a(n)=0.
By using the theorem and its corollary mentioned in the comments lines of the sequence A086766, we can prove that for m = 2, 3, ..., 275 a(10^m)=0.
What is the smallest odd prime p, such that (10^(p^2)-1)/(10^p-1) is a prime number (a(10^(p-1)) is nonzero)?
What is the smallest integer m, such that m > 1 and a(10^m) is nonzero?
Conjecture: If n is not of the form 10^m then a(n) is nonzero.
(End)

			a(2) = 2221 is a prime but 21 and 221 are composite.

Cf. A086766, A252491.

cp's OEIS Frontend

A086920 Smallest k such that n times concatenation of k with itself followed by a 1 is a prime, or 0 if no such number exists.

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A252491 a(n) = (10^(n^2) - 1)/(10^n - 1).

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PARI

A087403 a(n) = smallest prime of the form 10*K(n) + 1, where K is a number obtained by concatenation of n with itself, or 0 if no such prime exists.

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