|
%I #58 Jun 29 2019 11:26:02
%S 13,439,7069,27961,2864599,522134761,21107054541321649,
%T 12345678910987654321,1919434248892467772593071038679,
%U 24197857203266734883076090685781525281,1457624695486449811479514346937750581569993,1263023202979901596155544853826881857760357011832664659152364441
%N Primes whose base-b representation is the concatenation of the base-b representations of (1, 2, ..., k, k-1, ..., 1), for some b > 1 and some k > 1.
%C The sequences A[b] of numbers whose base-b representation is the concatenation of the base-b representations of (1, 2, ..., k, k-1, ..., 1), for a given b and all k >= 1, are recorded as A173427, A260853 - A260859, A173426, A260861 - A260866 and A260860 for bases b=2, ..., b=16 and b=60.
%C This is a supersequence of A260852, which lists only primes of the form A[b](b) - see A260343 for the b-values. In addition, the numbers A[b](b+2) are also prime for b=(2, 3, 11, 62, 182, ...), corresponding to terms a(3) = 7069, a(5) = 2864599, a(9) = 1919434248892467772593071038679, ... Still other examples are a(11) = A[12](16), a(12) = A[14](21), ... See the Broadhurst file for further data. [Edited by _N. J. A. Sloane_, Aug 24 2015]
%C Other subsequences of the form A[b](b+d) with at least 4 probable primes include: d=36, b=(2, 103, 117, 2804, ...); d=70, b=(74, 225, 229, 545, ...); d=200, b=(126, 315, 387, 2697, ...). For odd d, I know of 2 series with at least 3 probable primes: d=15, b=(18, 154, 1262, ...); d=165, b=(522, 602, 1858,...). - _David Broadhurst_, Aug 28 2015
%C See A261170 for the number of decimal digits of a(n); A261171 and A261172 for the k- and b-values such that a(n) = A[b](k). - _M. F. Hasler_, Sep 15 2015
%H David Broadhurst, <a href="/A260871/a260871_1.txt">Conjectured list of initial 434 terms</a> (The notation is that [15, [25, 29], 91] means that a(15) is A[25](29) with 91 decimal digits and [237, [895, 1289], 9933] means that a(237) is probably A[895](1289) with 9933 decimal digits.)
%e The first two terms are of the form A[b](b) with b=2 and b=3:
%e a(1) = 13 = 1101_2 = concat(1, 2=10_2, 1).
%e a(2) = 439 = 121021_3 = concat(1, 2, 3=10_3, 2, 1).
%e See comments for further examples.
%o (PARI) {L=1e99;A260871=List();for(b=2,9e9,for(n=b,9e9,if(L<p=sum(i=1,#n=concat(vector(n*2-1,k,digits(min(k,n*2-k),b))),n[i]*b^(#n-i)),break(2-(n>b)));ispseudoprime(p)&&listput(A260871,p)));vecsort(A260871)}
%Y The sequences A[b] are listed in A173427 for b=2, A260853 for b=3, A260854 for b=4, A260855 for b=5, A260856 for b=6, A260857 for b=7, A260858 for b=8, A260859 for b=9, A173426 for b=10, A260861 for b=11, A260862 for b=12, A260863 for b=13, A260864 for b=14, A260865 for b=15, A260866 for b=16, A260860 for b=60.
%Y Cf. A260343, A260852, A261408.
%K nonn,hard,base
%O 1,1
%A _M. F. Hasler_, Aug 02 2015; edited Aug 23 2015
|
