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A116430 The number of n-almost primes less than or equal to 10^n, starting with a(0)=1.

%I A116430 #20 Aug 23 2024 17:18:26
%S A116430 1,4,34,247,1712,11185,68963,409849,2367507,13377156,74342563,
%T A116430 407818620,2214357712,11926066887,63809981451,339576381990,
%U A116430 1799025041767,9494920297227,49950199374227,262036734664892
%N A116430 The number of n-almost primes less than or equal to 10^n, starting with a(0)=1.
%C A116430 If instead we asked for those less than or equal to 2^n, then the sequence is A000012.
%t A116430 AlmostPrimePi[k_Integer, n_] := Module[{a, i}, a[0] = 1; If[k == 1, PrimePi[n], Sum[PrimePi[n/Times @@ Prime[Array[a, k - 1]]] - a[k - 1] + 1, Evaluate[ Sequence @@ Table[{a[i], a[i - 1], PrimePi[(n/Times @@ Prime[Array[a, i - 1]])^(1/(k - i + 1))]}, {i, k - 1}]]]]]; (* _Eric W. Weisstein_, Feb 07 2006 *)
%t A116430 Table[ AlmostPrimePi[n, 10^n], {n, 0, 13}]
%o A116430 (PARI)
%o A116430 almost_prime_count(N, k) = if(k==1, return(primepi(N))); (f(m, p, k, j=0) = my(c=0, s=sqrtnint(N\m, k)); if(k==2, forprime(q=p, s, c += primepi(N\(m*q))-j; j += 1), forprime(q=p, s, c += f(m*q, q, k-1, j); j += 1)); c); f(1, 2, k);
%o A116430 a(n) = if(n == 0, 1, almost_prime_count(10^n, n)); \\ _Daniel Suteu_, Jul 10 2023
%o A116430 (Python)
%o A116430 from math import prod, isqrt
%o A116430 from sympy import primerange, integer_nthroot, primepi
%o A116430 def A116430(n):
%o A116430     if n<=1: return 3*n+1
%o A116430     def g(x,a,b,c,m): yield from (((d,) for d in enumerate(primerange(b,isqrt(x//c)+1),a)) if m==2 else (((a2,b2),)+d for a2,b2 in enumerate(primerange(b,integer_nthroot(x//c,m)[0]+1),a) for d in g(x,a2,b2,c*b2,m-1)))
%o A116430     return int(sum(primepi(10**n//prod(c[1] for c in a))-a[-1][0] for a in g(10**n,0,1,1,n))) # _Chai Wah Wu_, Aug 23 2024
%Y A116430 Cf. A036352, A114106, A114453.
%Y A116430 Cf. A078840, A078841, A078842, A116432, A078843, A116426, A078844, A116427, A078845, A116428, A116429, A116430, A078846, A116431.
%Y A116430 Cf. A006880, A036352, A109251, A114106, A114453, A120047 - A120053.
%K A116430 nonn
%O A116430 0,2
%A A116430 _Robert G. Wilson v_, Feb 10 2006, Jun 01 2006
%E A116430 Edited by _N. J. A. Sloane_, Aug 08 2008 at the suggestion of R. J. Mathar
%E A116430 a(15)-a(16) from _Donovan Johnson_, Oct 01 2010
%E A116430 a(17)-a(19) from _Daniel Suteu_, Jul 10 2023