cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

A215218 Number of sphenic numbers, i.e., numbers with exactly three distinct prime factors, up to 10^n.

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%I A215218 #26 Aug 26 2024 13:32:21
%S A215218 0,5,135,1800,19919,206964,2086746,20710806,203834084,1997171674,
%T A215218 19522428788,190614467420,1860310801454,18155356377267,
%U A215218 177224592578839,1730651760050923,16908343191198752,165279853754232019,1616504757072680964
%N A215218 Number of sphenic numbers, i.e., numbers with exactly three distinct prime factors, up to 10^n.
%H A215218 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 A215218 a(2) = 5 since there are the five sphenic numbers 30, 42, 66, 70, 78 up to 100.
%t A215218 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 A215218 (Python)
%o A215218 from math import isqrt
%o A215218 from sympy import primepi, primerange, integer_nthroot
%o A215218 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 A215218 Cf. A007304.
%K A215218 nonn
%O A215218 1,2
%A A215218 _Martin Renner_, Aug 06 2012
%E A215218 a(8)-a(19) from _Henri Lifchitz_, Nov 11 2012