A384249 The number of integers k from 1 to n such that the greatest divisor of k that is an infinitary divisor of n is squarefree.

1, 2, 3, 3, 5, 6, 7, 6, 8, 10, 11, 9, 13, 14, 15, 15, 17, 16, 19, 15, 21, 22, 23, 18, 24, 26, 24, 21, 29, 30, 31, 30, 33, 34, 35, 24, 37, 38, 39, 30, 41, 42, 43, 33, 40, 46, 47, 45, 48, 48, 51, 39, 53, 48, 55, 42, 57, 58, 59, 45, 61, 62, 56, 48, 65, 66, 67, 51

Offset

1,2

Links

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

Formula

Multiplicative with a(p) = p, and a(p^e) = p^e * (1 - 1/p^A065176(e)) for e >= 2.

Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = Product_{p prime} f(1/p) = 0.93444998595445071889..., and f(x) = 1 - (1-x^2) * Sum_{k>=2} x^(2^k)/(1-x^(2^k));

Mathematica

f[p_, e_] := p^e*(1 - 1/p^(2^IntegerExponent[e - Mod[e, 2], 2])); f[p_, 1] := p; 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, 2] == 1, 1, 1 - 1/f[i, 1]^(1 << valuation(f[i, 2] - f[i, 2]%2, 2)))); }

Crossrefs

Cf. A005117, A065176, A077609.

Analogous sequences: A063659, A384048.

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), this sequence (squarefree), A384250 (powerful), A384251 (odd), A384252 (power of 2).

Sequence in context: A359588 A337868 A063659 * A392877 A392878 A392879

Adjacent sequences: A384246 A384247 A384248 * A384250 A384251 A384252

Keyword

nonn,easy,mult

Author

Amiram Eldar, May 23 2025

Status

approved