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A260871
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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.
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7
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13, 439, 7069, 27961, 2864599, 522134761, 21107054541321649, 12345678910987654321, 1919434248892467772593071038679, 24197857203266734883076090685781525281, 1457624695486449811479514346937750581569993, 1263023202979901596155544853826881857760357011832664659152364441
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OFFSET
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1,1
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COMMENTS
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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.
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]
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
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LINKS
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David Broadhurst, Conjectured list of initial 434 terms (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.)
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EXAMPLE
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The first two terms are of the form A[b](b) with b=2 and b=3:
a(1) = 13 = 1101_2 = concat(1, 2=10_2, 1).
a(2) = 439 = 121021_3 = concat(1, 2, 3=10_3, 2, 1).
See comments for further examples.
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PROG
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(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)}
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CROSSREFS
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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.
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KEYWORD
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nonn,hard,base
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AUTHOR
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STATUS
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approved
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