Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).
%I #26 Aug 26 2024 13:32:21
%S 0,5,135,1800,19919,206964,2086746,20710806,203834084,1997171674,
%T 19522428788,190614467420,1860310801454,18155356377267,
%U 177224592578839,1730651760050923,16908343191198752,165279853754232019,1616504757072680964
%N Number of sphenic numbers, i.e., numbers with exactly three distinct prime factors, up to 10^n.
%H Paul Kinlaw, <a href="https://www.researchgate.net/publication/330506122_LOWER_BOUNDS_FOR_NUMBERS_WITH_THREE_PRIME_FACTORS">Lower bounds for numbers with three prime factors</a>, Husson University, Bangor, ME, 2019. Also in <a href="http://math.colgate.edu/~integers/t22/t22.Abstract.html">Integers</a> (2019) 19, Article #A22.
%e a(2) = 5 since there are the five sphenic numbers 30, 42, 66, 70, 78 up to 100.
%t f[n_] := Sum[ PrimePi[n/(Prime@ i*Prime@ j)] - j, {i, PrimePi[n^(1/3)]}, {j, i +1, PrimePi@ Sqrt[n/Prime@ i]}]; (* _Robert G. Wilson v_, Dec 28 2016 *)
%o (Python)
%o from math import isqrt
%o from sympy import primepi, primerange, integer_nthroot
%o def A215218(n): return int(sum(primepi(10**n//(k*m))-b for a,k in enumerate(primerange(integer_nthroot(10**n,3)[0]+1),1) for b,m in enumerate(primerange(k+1,isqrt(10**n//k)+1),a+1))) # _Chai Wah Wu_, Aug 26 2024
%Y Cf. A007304.
%K nonn
%O 1,2
%A _Martin Renner_, Aug 06 2012
%E a(8)-a(19) from _Henri Lifchitz_, Nov 11 2012