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.

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

Amiram Eldar, Table of n, a(n) for n = 1..10000

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

Cf. A000079, A006519, A077609.

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

Adjacent sequences: A384249 A384250 A384251 * A384253 A384254 A384255

Keyword

nonn,easy,mult

Author

Amiram Eldar, May 23 2025

Status

approved