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A003139
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Number of coprime chains with largest member n.
(Formerly M0129)
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2
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1, 1, 1, 1, 2, 1, 3, 1, 3, 2, 9, 1, 10, 2, 4, 3, 19, 1, 20, 2, 6, 4, 32, 1, 21, 7, 16, 7, 84, 1, 85, 9, 18, 11, 35, 3, 161, 15, 30, 6, 212, 2, 214, 15, 12, 19, 260, 3, 154, 11, 62, 31, 521, 5, 129, 19, 90, 54, 818, 2, 820, 54, 44, 57, 207, 7, 1189, 62, 147, 8, 1406
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listen;
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OFFSET
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1,5
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COMMENTS
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A coprime chain is a nonempty set of integers greater than 1 such that all primes <= the largest member divide exactly one term of the set. - Charles R Greathouse IV, Apr 24 2013
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REFERENCES
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N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
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Charlie Neder, Table of n, a(n) for n = 1..299
R. C. Entringer, The number of coprime chains with largest member n, Proc. Amer. Math. Soc., 16 (1965), 806-810.
R. C. Entringer, The number of coprime chains with largest member n, Proc. Amer. Math. Soc., 16 (1965), 806-810. [Annotated scanned copy]
R. C. Entringer, Some properties of certain sets of coprime integers, Proc. Amer. Math. Soc. 16 (1965), 515-521.
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FORMULA
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Entringer proves that log a(n) ~ sqrt(n). - Charles R Greathouse IV, Apr 24 2013
If p and q are consecutive primes with q > p, then a(q) = a(q-1) + a(q-2) + ... + a(p). - Charlie Neder, Dec 15 2018
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EXAMPLE
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The a(11) = 9 coprime chains with largest element 11 are {2,3,5,7,11}, {2,5,7,9,11}, {3,4,5,7,11}, {3,5,7,8,11}, {3,7,10,11}, {4,5,7,9,11}, {5,6,7,11}, {5,7,8,9,11}, and {7,9,10,11}. - Charlie Neder, Dec 15 2018
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PROG
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(PARI) isCC(v)=forprime(p=2, vecmax(v), if(sum(i=1, #v, v[i]%p==0)!=1, return(0))); 1
a(n)=my(v=vector(n-1, i, i+1)); sum(i=2^(n-2), 2^(n-1)-1, isCC(vecextract(v, i))) \\ Charles R Greathouse IV, Apr 24 2013
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CROSSREFS
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Cf. A003140.
Sequence in context: A224762 A039776 A048864 * A244797 A308659 A145652
Adjacent sequences: A003136 A003137 A003138 * A003140 A003141 A003142
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KEYWORD
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nonn,nice
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AUTHOR
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N. J. A. Sloane.
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EXTENSIONS
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a(56)-a(71) from Charlie Neder, Dec 15 2018
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STATUS
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approved
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