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A086766 a(n) = smallest r where (concatenation of n, r times with itself)*10 + 1 is a prime given by A087403(n), or 0 if no such number exists. 4
1, 3, 1, 1, 11, 1, 1, 2, 2, 1, 9, 3, 1, 5, 1, 3, 15, 1, 1, 2, 1, 60, 3, 1, 1, 2, 1, 1, 5, 5, 1, 2, 1, 6, 12, 3, 12, 3, 5, 1, 2, 1, 1, 5, 3, 1, 0, 2, 1, 9, 2, 1, 6, 1, 6, 18, 1, 3, 45, 1, 6, 3, 1, 1, 2, 1, 0, 3, 1, 1, 2, 3, 4, 8, 1, 1, 6, 2, 36, 96, 1, 1, 5, 304, 6, 2, 6, 1, 2, 2, 1, 2, 5, 1, 6, 5, 1, 2, 1, 0 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

Conjecture: No term is zero. [Warning: This is known to be wrong, see below. - M. F. Hasler, Jan 08 2015]

a(47), a(67), a(100), a(107), a(114) are zero or larger than 1000. - Ray Chandler, Sep 23 2003; edited by M. F. Hasler, Jan 08 2015

a(47) > 10000 or 0. a(67) > 10000 or 0. a(100) > 10000 or 0. a(107) = 2478. a(114) = 1164. See link for more details. - Derek Orr, Oct 02 2014

From Farideh Firoozbakht, Jan 07 2015: (Start)

The conjecture is not true and there exist many numbers n such that a(n)=0.

Theorem: If m is a positive integer and a(10^m)=r then r+1 divides m+1.

Corollary: If p is a prime number then a(10^(p-1))=0 or (10^(p^2)-1)/(10^p-1) is a prime number.

By using the theorem and its corollary 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 (and a(10^(p-1)) could be nonzero)?

What is the smallest integer m > 1 such that a(10^m) is nonzero?

Conjecture: If n is not of the form 10^m then a(n) is nonzero.

M. F. Hasler has checked proofs of the theorem and its corollary.

(End)

LINKS

Table of n, a(n) for n=1..100.

Derek Orr, Values of a(n) > 1000 for n < 1000

EXAMPLE

a(2) = 3, 2221 is a prime but 21 and 221 are composite.

PROG

(PARI)

a(n)=for(k=1, 10^4, if(ispseudoprime((n/(10^#Str(n)-1))*(10^(#Str(n)*k+1)-10)+1), return(k)))

vector(46, n, a(n)) \\ Derek Orr, Oct 02 2014

CROSSREFS

Cf. A087403.

Sequence in context: A060540 A087647 A100265 * A078688 A082466 A120270

Adjacent sequences:  A086763 A086764 A086765 * A086767 A086768 A086769

KEYWORD

base,nonn

AUTHOR

Amarnath Murthy, Sep 10 2003

EXTENSIONS

More terms from Ray Chandler, Sep 23 2003

STATUS

approved

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Last modified December 1 13:38 EST 2021. Contains 349429 sequences. (Running on oeis4.)