A365347 -id:A365347 - cp's OEIS frontend

A365346 The sum of divisors of the smallest square divisible by n.

1, 7, 13, 7, 31, 91, 57, 31, 13, 217, 133, 91, 183, 399, 403, 31, 307, 91, 381, 217, 741, 931, 553, 403, 31, 1281, 121, 399, 871, 2821, 993, 127, 1729, 2149, 1767, 91, 1407, 2667, 2379, 961, 1723, 5187, 1893, 931, 403, 3871, 2257, 403, 57, 217, 3991, 1281, 2863

Offset: 1

Amiram Eldar, Sep 02 2023

The number of these divisors is A365345(n).

The sum of divisors of the square root of the smallest square divisible by n is A365347(n).

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

Cf. A000203, A053143, A365345, A365347.

f[p_, e_] := (p^(e + 1 + Mod[e, 2]) - 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]^(f[i,2] + 1 + f[i,2]%2) - 1)/(f[i,1] - 1));}

a(n) = sigma(n*core(n)); \\ Michel Marcus, Sep 02 2023

a(n) = A000203(A053143(n)).
Multiplicative with a(p^e) = (p^(e + 1 + (e mod 2)) - 1)/(p - 1).
Dirichlet g.f.: zeta(s) * zeta(2*s-2) * Product_{p prime} (1 + 1/p^(s-2) + 1/p^(s-1) - 1/p^(2*s-2)).
Sum_{k=1..n} a(k) ~ c * n^3, where c = (Pi^2/45) * zeta(3) * Product_{p prime} (1 + 1/p^2 - 1/p^3) = 0.344306233314... .

A365481 The sum of unitary divisors of the smallest number whose square is divisible by n.

Original entry on oeis.org

1, 3, 4, 3, 6, 12, 8, 5, 4, 18, 12, 12, 14, 24, 24, 5, 18, 12, 20, 18, 32, 36, 24, 20, 6, 42, 10, 24, 30, 72, 32, 9, 48, 54, 48, 12, 38, 60, 56, 30, 42, 96, 44, 36, 24, 72, 48, 20, 8, 18, 72, 42, 54, 30, 72, 40, 80, 90, 60, 72, 62, 96, 32, 9, 84, 144, 68, 54, 96

Offset: 1

Amiram Eldar, Sep 05 2023

The number of unitary divisors of the smallest number whose square is divisible by n is the same as the number of unitary divisors of n, A034444(n).

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

Cf. A002117, A013661, A019554, A034444, A034448, A365347, A365479, A365480.

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

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

from math import prod
from sympy import factorint
def A365481(n): return prod(p**((e>>1)+(e&1))+1 for p,e in factorint(n).items()) # Chai Wah Wu, Sep 05 2023

a(n) = A034448(A019554(n)).
Multiplicative with a(p^e) = p^(ceiling(e/2)) + 1.
Dirichlet g.f.: zeta(s) * zeta(2*s-1) * Product_{p prime} (1 + 1/p^(s-1) - 1/p^(2*s-1) - 1/p^(3*s-1)).
Sum_{k=1..n} a(k) ~ c * n^2, where c = (1/2) * zeta(2) * zeta(3) * Product_{p prime} (1 - 1/p^2 - 1/p^3 + 1/p^4 - 1/p^5 + 1/p^6) = 0.515959523197... .

cp's OEIS Frontend

A365346 The sum of divisors of the smallest square divisible by n.

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A365481 The sum of unitary divisors of the smallest number whose square is divisible by n.

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Author

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Mathematica

PARI

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Formula