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URL: https://oeis.org/A384252

⇱ A384252 - OEIS

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A384252
The number of integers k from 1 to n such that the greatest divisor of k that is an infinitary divisor of n is a power of 2.
4
1, 2, 2, 4, 4, 4, 6, 8, 8, 8, 10, 8, 12, 12, 8, 16, 16, 16, 18, 16, 12, 20, 22, 16, 24, 24, 18, 24, 28, 16, 30, 32, 20, 32, 24, 32, 36, 36, 24, 32, 40, 24, 42, 40, 32, 44, 46, 32, 48, 48, 32, 48, 52, 36, 40, 48, 36, 56, 58, 32, 60, 60, 48, 64, 48, 40, 66, 64, 44
OFFSET
1,2
LINKS
FORMULA
Multiplicative with a(2^e) = 2^e, and a(p^e) = p^e * (1 - 1/p^A006519(e)) if p is an odd prime.
a(n) = A384247(n)/A384251(n).
Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = Product_{p prime >= 3} f(1/p) = 0.83603370570658499764..., and f(x) = 1 - (1-x)*Sum_{k>=1} x^(2^k)/(1-x^(2^k)).
MATHEMATICA
f[p_, e_] := p^e*(1 - 1/p^(2^(IntegerExponent[e, 2]))); f[2, e_] := 2^e; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]
PROG
(PARI) a(n) = {my(f = factor(n)); n * prod(i = 1, #f~, if(f[i, 1] == 2, 1, (1 - 1/f[i, 1]^(1 << valuation(f[i, 2], 2))))); }
CROSSREFS
Analogous sequences: A062570, A384056.
The number of integers k from 1 to n such that the greatest divisor of k that is an infinitary divisor of n is: A384247(1), A384249 (squarefree), A384250 (powerful), A384251 (odd), this sequence (power of 2).
Sequence in context: A062570 A108514 A317419 * A384056 A364843 A372678
KEYWORD
nonn,easy,mult
AUTHOR
Amiram Eldar, May 23 2025
STATUS
approved