A366743 -id:A366743 - cp's OEIS frontend

A366742 The number of divisors of the least coreful infinitary divisor of n.

1, 2, 2, 3, 2, 4, 2, 2, 3, 4, 2, 6, 2, 4, 4, 5, 2, 6, 2, 6, 4, 4, 2, 4, 3, 4, 2, 6, 2, 8, 2, 2, 4, 4, 4, 9, 2, 4, 4, 4, 2, 8, 2, 6, 6, 4, 2, 10, 3, 6, 4, 6, 2, 4, 4, 4, 4, 4, 2, 12, 2, 4, 6, 3, 4, 8, 2, 6, 4, 8, 2, 6, 2, 4, 6, 6, 4, 8, 2, 10, 5, 4, 2, 12, 4, 4

Offset: 1

Amiram Eldar, Oct 19 2023

The sum of divisors of the least coreful infinitary divisor of n is A366744(n).

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

Cf. A000005, A006519, A138302, A365296, A366743, A366744.

f[p_, e_] := 2^IntegerExponent[e, 2] + 1; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

a(n) = vecprod(apply(x -> 2^valuation(x, 2) + 1, factor(n)[, 2]));

a(n) = A000005(A365296(n)).
a(n) = A000005(n) if and only if n is in A138302.
Multiplicative with a(p^e) = A006519(e) + 1.

A366744 The sum of divisors of the least coreful infinitary divisor of n.

Original entry on oeis.org

1, 3, 4, 7, 6, 12, 8, 3, 13, 18, 12, 28, 14, 24, 24, 31, 18, 39, 20, 42, 32, 36, 24, 12, 31, 42, 4, 56, 30, 72, 32, 3, 48, 54, 48, 91, 38, 60, 56, 18, 42, 96, 44, 84, 78, 72, 48, 124, 57, 93, 72, 98, 54, 12, 72, 24, 80, 90, 60, 168, 62, 96, 104, 7, 84, 144, 68

Offset: 1

Amiram Eldar, Oct 19 2023

The number of divisors of the least coreful infinitary divisor of n is A366742(n).

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

Cf. A000203, A006519, A138302, A339597, A365296, A366742, A366743.

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

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

a(n) = A000203(A365296(n)).
a(n) = A000203(n) if and only if n is in A138302.
Multiplicative with a(p^e) = (p^(A006519(e)+1) - 1)/(p - 1).
Sum_{k=1..n} a(k) ~ c * n^2, where c = (1/2) * Product_{p prime} (1 - p/(p^2-1) + Sum_{e>=1} 1/p^f(e)) = 0.696427154..., where f(k) = 2*k - A006519(k) = A339597(k-1).

cp's OEIS Frontend

A366742 The number of divisors of the least coreful infinitary divisor of n.

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