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A282026
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a(n) is the smallest m with gcd(m, 2*n+1) = 1 such that 2*n + 2*m + 1 is composite.
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4
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4, 11, 2, 1, 8, 2, 1, 17, 2, 1, 2, 1, 1, 4, 2, 1, 1, 2, 1, 5, 2, 1, 2, 1, 1, 2, 1, 1, 4, 2, 1, 1, 2, 1, 4, 2, 1, 1, 2, 1, 2, 1, 1, 2, 1, 1, 1, 2, 1, 8, 2, 1, 8, 2, 1, 2, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 4, 2, 1, 1, 1, 1, 4, 2, 1, 1, 2, 1, 1, 2, 1, 2, 1, 1, 2, 1
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OFFSET
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0,1
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COMMENTS
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Starting at 2*n + 1, find the next odd composite number 2*n + 2*m + 1 that is relatively prime to 2*n + 1; then a(n) = m.
Since 2*n + 3 is relatively prime to 2*n + 1, and (2*n + 3)^2 is composite, a(n) <= 2*n^2 + 5*n + 4 (this is tight for n=0 and n=1).
Up to n = 10^7, a(n) are from the set [1, 2, 4, 5, 7, 8, 11, 13, 14, 16, 17, 19, 22]. First occurrence of 14 is a(99412), first occurrence of 22 is a(7225627). [Thanks to Altug Alkan for pointing out a(99412).]
(End)
Based on Altug Alkan's b-file, the records in this sequence are 4, 11, 17, 19, ... and occur at positions 1, 2, 8, 638, ... If the sequence is unbounded, then these two subsidiary sequences should be added to the OEIS (if they are new). - N. J. A. Sloane, Feb 13 2017
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LINKS
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EXAMPLE
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When n=1, 2*n + 1 = 3, and 5, 7, 9, 11, 13, 15, 17, 19, 21, 23 are all either prime or have a common factor with 3. The next term, 25, is OK, and so a(1) = (25 - 3)/2 = 11.
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MATHEMATICA
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Table[m = 1; While[Nand[CoprimeQ[m, 2 n + 1], CompositeQ[2 (n + m) + 1]], m++]; m, {n, 0, 120}] (* Michael De Vlieger, Feb 18 2017 *)
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PROG
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(PARI) a(n) = my(k=1); while(isprime(2*n+2*k+1) || gcd(2*n+1, k) != 1, k++); k; \\ Altug Alkan, Feb 13 2017
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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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STATUS
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approved
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