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Number of unitary prime factors of lcm[1..n], i.e., primes in LCM with exponent 1.
2

%I #16 Aug 03 2017 04:16:22

%S 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,

%T 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,

%U 13,14,14,14,14,14,14,15,15,15,15,16,16,17,17,17,17,17,17,18,18,18,18,19

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

%C Number of primes in the interval ]sqrt(n), n], (i.e., excluding sqrt(n) but including n). - _Lekraj Beedassy_, Mar 31 2005

%H G. C. Greubel, <a href="/A056812/b056812.txt">Table of n, a(n) for n = 1..1000</a>

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

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

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

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

%e 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.

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

%o (PARI) for(n=1,100, print1(primepi(n) - primepi(sqrt(n)), ", ")) \\ _G. C. Greubel_, May 13 2017

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

%K nonn

%O 1,3

%A _Labos Elemer_, Aug 28 2000