A174583 a(k) is the least n such that the concatenation (n - k)"n is a prime number, for k >= 0.

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, 77

Offset: 0

Author

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

Keywords

• base
• nonn

Comments

See comments and references for A174414.

10^d*(n - k) + n has to be prime for the least d-digit n > k (k >= 0).

For (n - k)"n to be a prime, n must end in the digit 1, 3, 7, or 9.

Conjecture: a(k) = a(k+1) for an infinite number of k's.

As n > k, the number of a(k) is finite, and can be easily bounded from above.

1, 11, ... appear only once in the sequence; 3, 9, 13, 19, 21, 23, ... appear twice; 7, 17, ... three times; and so on.

Does each n that ends in the digit 1, 3, 7, or 9 appear in this sequence?

Note this interesting observation that first occurs for k = 84: 9291013 = prime(620602) = (1013 - 84)"1013, a(84) = 1013. A second example is: 9381037 = prime(626219) = (1037 - 99)"1037.

Let k be a multiple of 7, 11, or 13, then no 3-digit n exists such that (n - k)"n is prime. Proof: 10^3*(n - k) + n = n * (10^3+1) - k * 10^3 = 7 * 11 * 13 * n - k * 10^3 is not prime, as k is a multiple of 7, 11, or 13.

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

Examples

			11 = prime(5) = (1 - 0)"1, thus a(0) = 1.
23 = prime(9) = (3 - 1)"3, thus a(1) = 3.
13 = prime(6) = (3 - 2)"3, thus a(2) = 3.
139 = prime(34) = (39 - 38)"39, thus a(38) = 39.
9109 = prime(1130) = (109 - 100)"109, thus a(100) = 109.
		

Crossrefs

Cf. A174414, A089712, A171154, A173976, A174031.

Programs

Maple

mycat := (k, n) -> parse(cat(convert(n - k, string), convert(n, string))):
sol := (k, n) -> isprime(mycat(k, n)):
a := proc(k) local n; for n from k + 1 while not sol(k, n) do od; n end:
seq(a(k), k = 0..71);  # Peter Luschny, Sep 20 2024

Extensions

Edited, offset set to 0 and a(71) corrected by Peter Luschny, Sep 20 2024