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A373943
a(n) is the cardinality of the set containing all rational numbers of the form 2 <= m/2^(bigomega(m) - 1) <= n.
1
0, 1, 2, 2, 4, 4, 6, 7, 7, 7, 10, 11, 13, 13, 13, 15, 18, 19, 21, 22, 22, 22, 24, 25, 27, 29, 30, 31, 34, 35, 36, 37, 38, 38, 40, 41, 43, 45, 47, 48, 49, 50, 54, 57, 57, 58, 61, 62, 63, 63, 63, 65, 66, 67, 67, 70, 71, 74, 75, 77, 79, 82, 82, 84, 86, 89, 91
OFFSET
1,3
FORMULA
a(n) = card{x | x = m/2^(bigomega(m)-1), x<=n}.
a(n) = pi_k(n * 2^(k - 1)), with pi_k(n) as the counting function for k-almost primes and k sufficiently large.
k needs to be at least max(1, floor(log(n/2)/(log(3)-log(2)))) and m = n * 2^(k - 1).
a(n) = A374022(n) + A000720(n).
a(2^n) = A052130(n-1).
EXAMPLE
a(10) = 7 = card{2, 3, 9/2, 5, 27/4, 7, 15/2}.
MATHEMATICA
z = 100;
k[n_] := Max[1, Floor[Log[3/2, n/2]]];
m[n_] := n 2^(k[n] - 1);
PrimePiK = Table[0, Floor[Log[2, m[z]]], m[z]];
For[i = 2, i <= m[z], i++, PrimePiK[[PrimeOmega[i], i]] = 1]
PrimePiK = Accumulate /@ PrimePiK;
a = Table[PrimePiK[[k[n], m[n]]], {n, z}] (*sequence*)
x = Union@Select[Table[i/2^(PrimeOmega[i] - 1), {i, 1, m[z], 2}], # <= z &] (*set*)
PROG
(PARI) nap(n, k) = sum(i=1, n, bigomega(i)==k);
a(n) = my(k=max(1, floor(log(n/2)/(log(3)-log(2))))); nap(n*2^(k-1), k); \\ Michel Marcus, Jun 27 2024
CROSSREFS
KEYWORD
nonn
AUTHOR
Friedjof Tellkamp, Jun 23 2024
STATUS
approved