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A260851
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a(n) in base n is the concatenation of the base n expansions of (1, 2, 3, ..., n-1, n, n-1, ..., 3, 2, 1).
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14
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1, 13, 439, 27961, 3034961, 522134761, 131870760799, 45954960939217, 21107054541321649, 12345678910987654321, 8954302429379707945271, 7883984846509322664831433, 8281481197999449959084458465, 10228838696316240496325238416281, 14674825961700306151086890240104831
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OFFSET
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1,2
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COMMENTS
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Sequences A173427, A260853 - A260859, A173426, A260861 - A260866, A260860 list the numbers A_b(n) whose base b expansion is the concatenation of the base b expansions of (1, 2, ..., n, n-1, ..., 1). For n < b these are the squares of the repdigits of length n in base b, so the first candidate for a prime is the term with n = b. These are the numbers listed here. Sequence A260343 gives the bases b for which this is indeed a prime, the corresponding primes a(A260343(n)) are listed in A260852.
The initial term a(1) = 1 refers to the unary or "tally mark" representation of the numbers, cf. A000042. It can be considered as purely conventional.
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LINKS
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N. J. A. Sloane, Table of n, a(n) for n = 1..100
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FORMULA
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a(n) = n*r + (r - n)*(1 + n*r) = (r - n + 1)*(1 + n*r) - 1, where r = (n^n-1)/(n-1) is the base n repunit of length n, r = 1 for n = 1.
Another closed-form expression for the series is a(n) = (n^(2*n+1) + (-n^3 + 2*n^2 - 2*n - 1)*n^n + 1)/(n - 1)^2. - Serge Batalov, Aug 02 2015
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EXAMPLE
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a(1) = 1 is the "concatenation" of (1) which is the unary representation of 1, cf A000042.
a(2) = 13 = 1101[2] = concatenation of (1, 10, 1), where 10 is the base 2 representation of 2.
a(3) = 439 = 121021[3] = concatenation of (1, 2, 10, 2, 1), where 10 is the base 3 representation of 3.
a(10) = 12345678910987654321 is the concatenation of (1, 2, 3, ..., 9, 10, 9, 8, ..., 2, 1); it is also a prime.
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MAPLE
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f:=proc(b) local i;
add((i+1)*b^i, i=0..b-2) + b^b + add(i*b^(2*b-i), i=1..b-1); end;
[seq(f(b), b=1..25)]; # N. J. A. Sloane, Sep 26 2015
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MATHEMATICA
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Join[{1}, Table[((n^n - 1)/(n - 1) - n + 1) (1 + n (n^n - 1)/(n - 1)) - 1, {n, 2, 30}]] (* Vincenzo Librandi, Aug 02 2015 *)
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PROG
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(PARI) A260851(n)=(1+n*r=if(n>2, n^n\(n-1), n*2-1))*(r-n+1)-1
(Magma) [1] cat [((n^n-1)/(n-1) - n + 1)*(1 + n*(n^n-1)/(n-1)) - 1: n in [2..15]]; // Vincenzo Librandi, Aug 02 2015
(Python) def A260851(n): return sum(i*(n**(2*n-i)+n**(i-1)) for i in range(1, n)) + n**n # Ya-Ping Lu, Dec 23 2021
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CROSSREFS
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Cf. A173427, A260853 - A260859, A173426, A260861 - A260866, A260860.
For primes in this sequence see A260343, A260852.
Sequence in context: A012084 A114759 A260871 * A260852 A012832 A102075
Adjacent sequences: A260848 A260849 A260850 * A260852 A260853 A260854
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KEYWORD
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nonn,base
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AUTHOR
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M. F. Hasler, Aug 01 2015
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STATUS
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approved
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