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

Author

Labos Elemer, Aug 28 2000

Keywords

• nonn

Comments

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

Examples

			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.
		

Links

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

Crossrefs

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

Programs

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 *)

PARI

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

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)).