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A056812
Number of unitary prime factors of lcm[1..n], i.e., primes in LCM with exponent 1.
2
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
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
KEYWORD
nonn
AUTHOR
Labos Elemer, Aug 28 2000
STATUS
approved