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

%I A174583 #13 Sep 20 2024 05:47:49
%S A174583 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,
%T A174583 27,27,29,31,33,31,33,39,47,37,39,37,39,39,41,51,47,47,61,47,49,49,53,
%U A174583 49,51,51,59,57,57,61,57,57,61,63,63,71,63,63,67,67,77,67,69,77,71,73,77
%N A174583 a(k) is the least n such that the concatenation (n - k)"n is a prime number, for k >= 0.
%C A174583 See comments and references for A174414.
%C A174583 10^d*(n - k) + n has to be prime for the least d-digit n > k (k >= 0).
%C A174583 For (n - k)"n to be a prime, n must end in the digit 1, 3, 7, or 9.
%C A174583 Conjecture: a(k) = a(k+1) for an infinite number of k's.
%C A174583 As n > k, the number of a(k) is finite, and can be easily bounded from above.
%C A174583 1, 11, ... appear only once in the sequence; 3, 9, 13, 19, 21, 23, ... appear twice; 7, 17, ... three times; and so on.
%C A174583 Does each n that ends in the digit 1, 3, 7, or 9 appear in this sequence?
%C A174583 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.
%C A174583 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.
%C A174583 Similar for k-digit n with given divisors and k > 3: 10^4 + 1 = 73 * 137, 10^5 + 1 = 11 * 9091.
%e A174583 11 = prime(5) = (1 - 0)"1, thus a(0) = 1.
%e A174583 23 = prime(9) = (3 - 1)"3, thus a(1) = 3.
%e A174583 13 = prime(6) = (3 - 2)"3, thus a(2) = 3.
%e A174583 139 = prime(34) = (39 - 38)"39, thus a(38) = 39.
%e A174583 9109 = prime(1130) = (109 - 100)"109, thus a(100) = 109.
%p A174583 mycat := (k, n) -> parse(cat(convert(n - k, string), convert(n, string))):
%p A174583 sol := (k, n) -> isprime(mycat(k, n)):
%p A174583 a := proc(k) local n; for n from k + 1 while not sol(k, n) do od; n end:
%p A174583 seq(a(k), k = 0..71);  # _Peter Luschny_, Sep 20 2024
%Y A174583 Cf. A174414, A089712, A171154, A173976, A174031.
%K A174583 base,nonn
%O A174583 0,2
%A A174583 Ulrich Krug (leuchtfeuer37(AT)gmx.de), Mar 23 2010
%E A174583 Edited, offset set to 0 and a(71) corrected by _Peter Luschny_, Sep 20 2024