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A247024
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Primes of the form x*(2^x - x) - 1.
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3
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OFFSET
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1,1
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COMMENTS
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Subsequence of A245014. a(n) represents the identity between (p + 4k^2 + 1) and (2n*4^k) for the least prime p defining A245014(k). Those first k where it occurs are: 1,2,3,5,41,42,62,183,357,407.
Consider the table of Stirling numbers of the second kind (A008277). The second column contains the numbers A000225, or 2^m - 1, and the first subdiagonal contains the triangular numbers. If a number appears in both sequences, we have the equation f(x) = 2^x - x^2 + x - 2 = 0 which has integer roots x = 1, 2, 3. Set g(x) = (x - 1)*(2^x - 1). Then it is found that the sum f(x) + g(x) for some even x defines this sequence and satisfies in common with A245014: Both sequences have three consecutive terms (those first) such that when they are represented in decimal the third term is the concatenation of the two terms preceding it.
Prime or PRP for x = 2, 4, 6, 10, 82, 84, 124, 366, 714, 814, 1584, 8938, 17812, 27054, 35380, 71358. - Jens Kruse Andersen, Sep 10 2014
The complete solution to the remark on Stirling2 numbers in a comment above is given in A076046. See also my Oct 08 2014 remark in the history. - Wolfdieter Lang, Oct 16 2014
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LINKS
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MAPLE
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MATHEMATICA
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Select[Table[n(2^n - n) - 1, {n, 200}], PrimeQ] (* Alonso del Arte, Sep 09 2014 *)
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PROG
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(PARI) listToVec(l:list, Comp, Omega=#l)=vector(Omega, i, l[i][Comp]);
fplusg(x)=x*(2^x-x)-1;
find_next_N_terms(N, resumeFrom:list=List(), verbose=0)= {
my(s:list, q, j, F, N0);
s=resumeFrom;
q=#s;
N0=q;
if(q, j=s[q][1]);
while(q<N0+N,
j++;
F=fplusg(2*j);
if(isprime(F), q++; listput(s, [j, F]); if(verbose, print1(", ("j")")), if(verbose, print1(", "j)))
);
s
} /* This will keep a record of the terms and the x values
generating these primes through fplusg(2*x); For example use:
listToVec(find_next_N_terms(11), 2);
in order to find the first 11 terms. */
(PFGW) ABC2 $a*(2^$a-$a)-1
a: from 2 to 100000 step 2
(Magma) [a: x in [0..450] | IsPrime(a) where a is x*(2^x - x)-1]; // Vincenzo Librandi, Oct 16 2014
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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Edited: 'penultimate column' replaced by 'first subdiagonal'. - Wolfdieter Lang, Oct 16 2014
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STATUS
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approved
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