A384248 The sum of the integers from 1 to n whose largest divisor that is an infinitary divisor of n is 1.

1, 1, 3, 6, 10, 6, 21, 16, 36, 20, 55, 36, 78, 42, 60, 120, 136, 72, 171, 120, 126, 110, 253, 96, 300, 156, 243, 252, 406, 120, 465, 256, 330, 272, 420, 432, 666, 342, 468, 320, 820, 252, 903, 660, 720, 506, 1081, 720, 1176, 600, 816, 936, 1378, 486, 1100, 672

Offset: 1

Author

Amiram Eldar, May 23 2025

Keywords

• nonn
• easy

Links

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

Crossrefs

Row sums of A384246.

Cf. A138302, A268335, A384247.

Analogous sequences: A023896, A200723, A333576.

Programs

Mathematica

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

PARI

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

Formula

a(n) = n * A384247(n) / 2, for n >= 2.

a(n) <= A333576(n), with equality if and only if n is in A138302.

a(n) >= A023896(n), with equality if and only if n is an exponentially odd number (A268335).

Sum_{k=1..n} a(k) ~ c * n^2 / 6, 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)).