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A003139 Number of coprime chains with largest member n.
(Formerly M0129)
2
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,5
COMMENTS
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
REFERENCES
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
LINKS
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.
FORMULA
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
EXAMPLE
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
PROG
(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
CROSSREFS
Cf. A003140.
Sequence in context: A224762 A039776 A048864 * A349918 A244797 A308659
KEYWORD
nonn,nice
AUTHOR
EXTENSIONS
a(56)-a(71) from Charlie Neder, Dec 15 2018
STATUS
approved

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Last modified March 18 22:56 EDT 2024. Contains 370952 sequences. (Running on oeis4.)