A116431 - OEIS

%I #17 Sep 28 2024 07:39:07

%S 1,5,48,434,3695,29165,218283,1569995,10950776,74621972,499495257,

%T 3297443264,21533211312,139411685398,896352197825,5730605551626,

%U 36465861350230

%N The number of n-almost primes less than or equal to 12^n, starting with a(0)=1.

%t 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 Table[ AlmostPrimePi[n, 12^n], {n, 12}]

%o (PARI)

%o 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 a(n) = if(n == 0, 1, almost_prime_count(12^n, n)); \\ _Daniel Suteu_, Jul 10 2023

%o (Python)

%o from math import isqrt, prod

%o from sympy import primerange, integer_nthroot, primepi

%o def A116431(n):

%o     if n<=1: return 4*n+1

%o     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     return int(sum(primepi(12**n//prod(c[1] for c in a))-a[-1][0] for a in g(12**n,0,1,1,n))) # _Chai Wah Wu_, Sep 28 2024

%Y Cf. A078840, A078841, A078842, A116432, A078843, A116426, A078844, A116427, A078845, A116428, A116429, A116430, A078846, A116431.

%K nonn,more,changed

%O 0,2

%A _Robert G. Wilson v_, Feb 10 2006

%E a(13)-a(14) from _Donovan Johnson_, Oct 01 2010

%E a(15)-a(16) from _Daniel Suteu_, Jul 10 2023