A261171 -id:A261171 - cp's OEIS frontend

A261172 Value of b for which A260871(n) = A[b](k), with k = A261171(n); A[b](k) = the number whose base-b representation is the concatenation of the base-b representations of (1, ..., k, k-1, ..., 1).

2, 3, 2, 4, 3, 6, 9, 10, 11, 16, 12, 14, 22, 18, 25, 20, 2, 6, 18, 14, 7, 40, 31, 25, 23, 20, 22, 62, 65, 68, 29, 23, 38, 26, 104, 6, 34, 52

Offset: 1

M. F. Hasler, Aug 23 2015

For more data, see the 2nd column of D. Broadhurst's list of [n, b, k, length(A260871(n))] given in A260871.

			A260871(1) = A[2](2), therefore a(1) = 2.
A260871(2) = A[3](3), therefore a(2) = 3.
A260871(3) = A[2](4), therefore a(3) = 2.

Cf. A173427, A260853 - A260859, A173426, A260861 - A260866 and A260860 for A[b] with b=2, ..., b=16 and b=60.

See also A260852 = { primes of the form A260851(b) = A[b](b), b in A260343 }.

A261172_list(LIM=1e499)={my(A=List(),p,d);for(b=2,9e9,for(n=b,9e9,if(LIMb)));ispseudoprime(p)&&listput(A,[log(p),n])));apply(t->t[2],vecsort(A))}

A260871(n) = A[a(n)](A261171(n)), where A[b](k) = Sum_{i=1..#d} d[i]*b^(#d-i), d = concatenation of (1, 2, ..., k, k-1, ..., 1) all written in base b.

A260871 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.

Original entry on oeis.org

13, 439, 7069, 27961, 2864599, 522134761, 21107054541321649, 12345678910987654321, 1919434248892467772593071038679, 24197857203266734883076090685781525281, 1457624695486449811479514346937750581569993, 1263023202979901596155544853826881857760357011832664659152364441

Offset: 1

M. F. Hasler, Aug 02 2015; edited Aug 23 2015

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
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

			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.

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.)

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.

Cf. A260343, A260852, A261408.

{L=1e99;A260871=List();for(b=2,9e9,for(n=b,9e9,if(Lb)));ispseudoprime(p)&&listput(A260871,p)));vecsort(A260871)}

A261170 Number of decimal digits of A260871(n), where A260871 lists primes whose base-b representation is the concatenation of the base-b representations of (1, 2, ..., k, k-1, ..., 1).

Original entry on oeis.org

2, 3, 4, 5, 7, 9, 17, 20, 31, 38, 43, 64, 64, 70, 91, 93, 102, 117, 120, 123, 127, 127, 127, 136, 160, 166, 176, 235, 321, 351, 353, 389, 403, 418, 418, 421, 422, 466, 542, 578, 579, 703, 706, 725, 731, 765, 780, 792, 795, 799, 803, 839, 840, 848, 849, 863

Offset: 1

M. F. Hasler, Sep 15 2015

Larger values based on computations by D. Broadhurst, cf. data file in A260871.

See A261171 and A261172 for the k- and b-values such that A260871(n) = A[b](k), where 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) are listed in A173427, A260853 - A260859, A173426, A260861 - A260866 and A260860 for bases b=2, ..., b=16 and b=60.

M. F. Hasler, Table of n, a(n) for n = 1..434

A261408 Numbers k such that A260864(k) is a prime or probable prime, where A260864(k) is the number whose base-14 representation is the concatenation of the base-14 representations of (1, ..., k, k-1, ..., 1).

Original entry on oeis.org

21, 34, 192, 310, 373, 536, 12000

Offset: 1

N. J. A. Sloane, Aug 22 2015

The subsequence of those A261171(n) for which A261172(n) = 14. - M. F. Hasler, Aug 23 2015

David Broadhurst, email to N. J. A. Sloane, Aug 23 2015

Cf. A260343, A260864.

See also A260871 which lists all primes of the form A[b](k), and A261171 and A261172 which list the k and b values of these primes. - M. F. Hasler, Aug 23 2015

More explicit title from M. F. Hasler, Aug 23 2015

cp's OEIS Frontend

A261172 Value of b for which A260871(n) = A[b](k), with k = A261171(n); A[b](k) = the number whose base-b representation is the concatenation of the base-b representations of (1, ..., k, k-1, ..., 1).

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A260871 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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A261170 Number of decimal digits of A260871(n), where A260871 lists primes whose base-b representation is the concatenation of the base-b representations of (1, 2, ..., k, k-1, ..., 1).

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A261408 Numbers k such that A260864(k) is a prime or probable prime, where A260864(k) is the number whose base-14 representation is the concatenation of the base-14 representations of (1, ..., k, k-1, ..., 1).

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