A003139 Number of coprime chains with largest member n. (Formerly M0129)

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

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), 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

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