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%I #14 Jul 16 2021 11:34:43
%S 4,6,8,9,26,1752
%N Composite numbers k such that the concatenation of all nonprime positive integers up to k in decreasing order is prime.
%C The terms of this sequence are composite terms of the sequence A099070 with the same order. Next term is greater than 6000 and the prime corresponding to the next term has more than 20000 digits. Number of digits of primes corresponding to the six known terms of the sequence are respectively 2, 3, 4, 5, 29, and 5010.
%H Carlos Rivera, <a href="http://www.primepuzzles.net/puzzles/puzz_008.htm">Puzzle 8. Primes by Listing</a>, The Prime Puzzles and Problems Connection.
%e 26 is a term: 26 is composite; nonprimes up to 26 are 1, 4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 22, 24, 25, 26; and 26252422212018161514121098641 is prime.
%t Do[If[ !PrimeQ[n]&&PrimeQ[(v={};Do[If[ !PrimeQ[n+1-j], v=Join[v, IntegerDigits[n+1-j]]], {j, n}];FromDigits[v])], Print[n]], {n, 6013}]
%t cnpQ[n_]:=PrimeQ[FromDigits[Flatten[IntegerDigits/@Select[Range[n,1,-1],!PrimeQ[#]&]]]]; Select[Range[1800],!PrimeQ[#]&&cnpQ[#]&] (* _Harvey P. Dale_, Jul 19 2020 *)
%Y Cf. A099070, A100003, A046284.
%K base,more,nonn,nice
%O 1,1
%A _Farideh Firoozbakht_, Nov 06 2004
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