

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
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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

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), 806810.
R. C. Entringer, The number of coprime chains with largest member n, Proc. Amer. Math. Soc., 16 (1965), 806810. [Annotated scanned copy]
R. C. Entringer, Some properties of certain sets of coprime integers, Proc. Amer. Math. Soc. 16 (1965), 515521.


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(q1) + a(q2) + ... + 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(n1, i, i+1)); sum(i=2^(n2), 2^(n1)1, isCC(vecextract(v, i))) \\ Charles R Greathouse IV, Apr 24 2013


CROSSREFS

Cf. A003140.
Sequence in context: A224762 A039776 A048864 * A244797 A308659 A145652
Adjacent sequences: A003136 A003137 A003138 * A003140 A003141 A003142


KEYWORD

nonn,nice


AUTHOR

N. J. A. Sloane.


EXTENSIONS

a(56)a(71) from Charlie Neder, Dec 15 2018


STATUS

approved



