A384247 The number of integers from 1 to n whose largest divisor that is an infinitary divisor of n is 1.

1, 1, 2, 3, 4, 2, 6, 4, 8, 4, 10, 6, 12, 6, 8, 15, 16, 8, 18, 12, 12, 10, 22, 8, 24, 12, 18, 18, 28, 8, 30, 16, 20, 16, 24, 24, 36, 18, 24, 16, 40, 12, 42, 30, 32, 22, 46, 30, 48, 24, 32, 36, 52, 18, 40, 24, 36, 28, 58, 24, 60, 30, 48, 48, 48, 20, 66, 48, 44, 24

Offset: 1

Author

Amiram Eldar, May 23 2025

Keywords

• nonn
• easy
• mult

Comments

Analogous to A047994, as A064380 is analogous to A116550.

Links

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

Crossrefs

Row lengths of A384246.

Cf. A000010, A001146, A006519, A047994, A064380, A091732, A116550, A138302, A268335, A384248.

Programs

Mathematica

f[p_, e_] := p^e*(1 - 1/p^(2^(IntegerExponent[e, 2]))); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

PARI

a(n) = {my(f = factor(n)); n * prod(i = 1, #f~, (1 - 1/f[i,1]^(1 << valuation(f[i,2], 2))));}

Formula

Multiplicative with a(p^e) = p^e * (1 - 1/p^A006519(e)).

a(n) >= A091732(n), with equality if and only if n is in A138302.

a(n) <= A047994(n), with equality if and only if n is in A138302.

a(n) >= A000010(n), with equality if and only if n is an exponentially odd number (A268335).

a(n) is odd if and only if n = 1 or 2^(2^k) for k >= 0 (A001146). a(2^(2^k)) = 2^(2^k)-1.

Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = Product_{p prime} f(1/p) = 0.66718130416373472394..., and f(x) = 1 - (1-x)*Sum_{k>=1} x^(2^k)/(1-x^(2^k)).