A384247 -id:A384247 - cp's OEIS frontend

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

Amiram Eldar, May 23 2025

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

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).

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]

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)))));}

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)).

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

Original entry on oeis.org

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

Amiram Eldar, May 23 2025

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

Row sums of A384246.
Cf. A138302, A268335, A384247.
Analogous sequences: A023896, A200723, A333576.

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]

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);

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)).

cp's OEIS Frontend

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.

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