A056812 Number of unitary prime factors of lcm[1..n], i.e., primes in LCM with exponent 1.

0, 1, 2, 1, 2, 2, 3, 3, 2, 2, 3, 3, 4, 4, 4, 4, 5, 5, 6, 6, 6, 6, 7, 7, 6, 6, 6, 6, 7, 7, 8, 8, 8, 8, 8, 8, 9, 9, 9, 9, 10, 10, 11, 11, 11, 11, 12, 12, 11, 11, 11, 11, 12, 12, 12, 12, 12, 12, 13, 13, 14, 14, 14, 14, 14, 14, 15, 15, 15, 15, 16, 16, 17, 17, 17, 17, 17, 17, 18, 18, 18, 18, 19

Offset

1,3

Comments

Number of primes in the interval ]sqrt(n), n], (i.e., excluding sqrt(n) but including n). - Lekraj Beedassy, Mar 31 2005

Links

G. C. Greubel, Table of n, a(n) for n = 1..1000

Formula

a(n) = A056169(A003418(n)).

a(n) = primepi(n) - primepi(sqrt(n)).

a(n) = A000720(n) - primepi(sqrt(n)).

a(n) = A001221(A003418(n)) - A000720(A000196(n)).

Example

n=100, lcm(100) has 25 prime factors of which only 2 and 3 have exponent larger than 1; resulting powers: 64 and 81. So 23 prime factors are unitary, i.e., with exponent 1, so a(100)=23.

Mathematica

Join[{0}, Table[Count[Transpose[FactorInteger[Product[Cyclotomic[k, 1], {k, 2, n}]]][[2]], 1], {n, 2, 100}]] (* G. C. Greubel, May 13 2017 *)

Prog

(PARI) for(n=1, 100, print1(primepi(n) - primepi(sqrt(n)), ", ")) \\ G. C. Greubel, May 13 2017

Crossrefs

Cf. A000196, A000720, A001221, A003418, A056169.

Sequence in context: A319447 A224961 A358023 * A210721 A082533 A230149

Adjacent sequences: A056809 A056810 A056811 * A056813 A056814 A056815

Keyword

nonn

Author

Labos Elemer, Aug 28 2000

Status

approved