A366742 -id:A366742 - cp's OEIS frontend

A366743 The sum of infinitary divisors of the least coreful infinitary divisor of n.

1, 3, 4, 5, 6, 12, 8, 3, 10, 18, 12, 20, 14, 24, 24, 17, 18, 30, 20, 30, 32, 36, 24, 12, 26, 42, 4, 40, 30, 72, 32, 3, 48, 54, 48, 50, 38, 60, 56, 18, 42, 96, 44, 60, 60, 72, 48, 68, 50, 78, 72, 70, 54, 12, 72, 24, 80, 90, 60, 120, 62, 96, 80, 5, 84, 144, 68, 90

Offset: 1

Amiram Eldar, Oct 19 2023

Also, the sum of unitary divisors of the least coreful infinitary divisor of n, A365296(n), since A365296(n) is a term of A138302, which is also the sequence of numbers whose sets of unitary divisors (A077610) and infinitary divisors (A077609) coincide.

The number of infinitary divisors of the least coreful infinitary divisor of n is A034444(n).

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

Cf. A000203, A005117, A006519, A034444, A034448, A049417, A077609, A077610, A138302, A339597, A365296, A366742, A366744.

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

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

a(n) = A034448(A365296(n)).
a(n) = A049417(A365296(n)).
a(n) = A000203(n) if and only if n is squarefree (A005117).
Multiplicative with a(p^e) = p^A006519(e) + 1.
Sum_{k=1..n} a(k) ~ c * n^2, where c = (1/2) * Product_{p prime} (1 - 1/(p+1) + Sum_{e>=1} 1/p^f(e)-1/p^(f(e)+1)) = 0.61865169..., where f(k) = 2*k - A006519(k) = A339597(k-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

A366743 The sum of infinitary divisors of the least coreful infinitary divisor of n.

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