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A120049
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Number of 8-almost primes less than or equal to 10^n.
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12
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0, 0, 0, 7, 105, 1418, 17572, 207207, 2367507, 26483012, 291646797, 3173159326, 34192782745, 365561221293, 3882841742380
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OFFSET
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0,4
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LINKS
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EXAMPLE
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There are 7 eight-almost primes up to 1000: 256, 384, 576, 640, 864, 896 & 960.
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MATHEMATICA
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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 *)
Table[AlmostPrimePi[8, 10^n], {n, 12}]
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PROG
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(Python)
from math import prod, isqrt
from sympy import primerange, integer_nthroot, primepi
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)))
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, 8))) # Chai Wah Wu, Aug 23 2024
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CROSSREFS
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Cf. A006880, A036352, A109251, A114106, A114453, A120047, A120048, A120049, A120050, A120051, A120052, A120053, A116430.
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KEYWORD
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nonn,changed
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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