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A174583 Smallest natural n = n(k) that concatenation (n-k)//n (k = 0, 1, 2, ...) is a prime number. 1
1, 3, 3, 7, 7, 17, 7, 9, 9, 11, 13, 17, 13, 19, 17, 19, 21, 21, 23, 27, 27, 23, 43, 33, 41, 27, 27, 29, 31, 33, 31, 33, 39, 47, 37, 39, 37, 39, 39, 41, 51, 47, 47, 61, 47, 49, 49, 53, 49, 51, 51, 59, 57, 57, 61, 57, 57, 61, 63, 63, 71, 63, 63, 67, 67, 77, 67, 69, 77, 71, 73, 71 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

See comments and references for A174414

10^d*(n-k) + n has to be a prime for the smallest d-digit natural n > k (k = 0, 1, 2, ...)

For (n-k)//n to be a prime n of course has necessarily end digit 1, 3, 7 or 9

It is conjectured that n(k) = n(k+1) for an infinite number of k's

As n > k the number N of a "possible" n(k) is finite, it is simple to give an upper bound for N

1, 11, ... appear only one time; 3, 9, 13, 19, 21, 23, ...: 2 times; 7, 17: 3 times; ...

Does EACH "possible" candidate n (naturals with end digit 1, 3, 7 or 9) "appear" in the sequence?

Note this interesting "phenomenon", first "occurrence" for case k = 84:

9291013 = prime(620602) = (1013-84)//1013, n(84) = 1013

2nd such: 9381037 = prime(626219) = (1037-99)//1037

Let k be a multiple of 7, 11 or 13 => no 3-digit n with property (n-k)//n a prime exists

Proof: 10^3*(n - k) + n = n * (10^3+1) - k * 10^3 = 7 * 11 * 13 * n - k * 10^3,

not prime as k is a multiple of 7, 11 or 13

Same "phenomenon" for k-digit n with appointed divisors and k > 3: 10^4+1 = 73 * 137, 10^5+1 = 11 * 9091, ...

LINKS

Table of n, a(n) for n=1..72.

EXAMPLE

11 = prime(5) = (1-0)//1, n(0) = 1

23 = prime(9) = (3-1)//3, n(1) = 3

13 = prime(6) = (3-2)//3, n(2) = 3

139 = prime(34) = (39-38)//39, n(38) = 39

9109 = prime(1130) = (109-100)//109, n(100) = 109

CROSSREFS

A089712, A171154, A173976, A174031, A174414

Sequence in context: A146450 A233810 A263869 * A226781 A147144 A152113

Adjacent sequences:  A174580 A174581 A174582 * A174584 A174585 A174586

KEYWORD

base,nonn,uned

AUTHOR

Ulrich Krug (leuchtfeuer37(AT)gmx.de), Mar 23 2010

STATUS

approved

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Last modified September 27 22:01 EDT 2022. Contains 357063 sequences. (Running on oeis4.)