|
%I #17 Jan 01 2020 17:51:17
%S 10243,12043,20143,20341,20431,23041,24103,30241,32401,40123,40213,
%T 40231,41023,41203,42013,43201,10235647,10236547,10243567,10243657,
%U 10245637,10247563,10254367,10254763,10256347,10256473,10257463,10264357
%N Primes whose digits are a permutation of (0, ..., m) for some m.
%C Starts with the 5-digit terms listed in A109176: There is no smaller prime of that form, since there is no odd number of that form with less than 3 digits, and those with digits {0,1,2} and {0,1,2,3} (as, e.g., 2013...) are all divisible by 3, thus composite.
%C For similar reasons, there cannot be terms with the 6 digits {0, ..., 5} or the 7 digits {0, ..., 6} (since 1 + ... + 5 (+ 6) is divisible by 3).
%C The 8-digit terms a(17)..a(2684) are also listed in A109177 and A109178 (in reverse order). Again, there are no 9- or 10-digit terms (since 0+1+...+8(+9) is divisible by 9). Therefore, the sequence has no terms beyond the 2684 terms < 76543210 listed in the b-file.
%H M. F. Hasler, <a href="/A187796/b187796.txt">Table of n, a(n) for n = 1..2684</a>
%F This sequence A187796 = A109176 union A109177.
%e As explained in the comments, there cannot be a term with fewer than 5 digits. The smallest number whose digits are a permutation of (0, ..., 4) is 10234, but this is even and cannot be a prime. The next larger one happens to be prime, so that's a(1) = 10243.
%e It is also explained in the comments why there's no term larger than 76543210. The largest odd numbers of the given form below this limit are of the form 7654xyz1 and 7654abc3, with xyz resp. abc permutations of 023 resp. 012. It happens that the case xyz=023 is the only one which yields a prime: this is the largest term of this sequence, a(2684) = 76540231 = A109178(1).
%t Select[Prime@ Range[10^6], {1} == Union@ Prepend[Differences@ #, 1 + First@ #] &@ Sort@ IntegerDigits@ # &] (* _Michael De Vlieger_, Aug 20 2017 *)
%t Table[Select[FromDigits /@ Permutations[Range[0, n]], PrimeQ[ #] && DigitCount[ #, 10, 0] == 1 &], {n, 9}] // Flatten (* _Harvey P. Dale_, Jan 01 2020 *)
%o (PARI) forprime(p=2,,#vecsort(t=digits(p),,8)==#t && #t==vecmax(t)+1 && print1(p","))
%K nonn,base,nice,fini,full
%O 1,1
%A _M. F. Hasler_, Jan 06 2013
|
