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Revision History for A365347

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A365347

The sum of divisors of the smallest number whose square is divisible by n.
(history; published version)

#9 by OEIS Server at Sun Sep 03 00:03:16 EDT 2023

LINKS

Amiram Eldar, <a href="/A365347/b365347_1.txt">Table of n, a(n) for n = 1..10000</a>

#8 by Michael De Vlieger at Sun Sep 03 00:03:16 EDT 2023

STATUS

reviewed

approved

Discussion

Sun Sep 03

00:03

OEIS Server: Installed first b-file as b365347.txt.

#7 by Andrew Howroyd at Sat Sep 02 23:09:35 EDT 2023

STATUS

proposed

reviewed

#6 by Michel Marcus at Sat Sep 02 08:49:20 EDT 2023

STATUS

editing

proposed

#5 by Michel Marcus at Sat Sep 02 08:49:17 EDT 2023

PROG

(PARI) a(n) = sigma(n/core(n, 1)[2]); \\ Michel Marcus, Sep 02 2023

STATUS

proposed

editing

#4 by Amiram Eldar at Sat Sep 02 06:00:54 EDT 2023

STATUS

editing

proposed

#3 by Amiram Eldar at Sat Sep 02 05:52:45 EDT 2023

LINKS

Amiram Eldar, <a href="/A365347/b365347_1.txt">Table of n, a(n) for n = 1..10000</a>

#2 by Amiram Eldar at Sat Sep 02 05:46:46 EDT 2023

NAME

allocated for Amiram EldarThe sum of divisors of the smallest number whose square is divisible by n.

DATA

1, 3, 4, 3, 6, 12, 8, 7, 4, 18, 12, 12, 14, 24, 24, 7, 18, 12, 20, 18, 32, 36, 24, 28, 6, 42, 13, 24, 30, 72, 32, 15, 48, 54, 48, 12, 38, 60, 56, 42, 42, 96, 44, 36, 24, 72, 48, 28, 8, 18, 72, 42, 54, 39, 72, 56, 80, 90, 60, 72, 62, 96, 32, 15, 84, 144, 68, 54

OFFSET

1,2

COMMENTS

The number of divisors of the smallest number whose square is divisible by n is A322483(n).

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

FORMULA

a(n) = A000203(A019554(n)).

Multiplicative with a(p^e) = (p^(e + 1 + (e mod 2)) - 1)/(p - 1).

Sum_{k=1..n} a(k) ~ c * n^2, where c = (1/2) * zeta(3) * Product_{p prime} (1 - 1/(p^2*(p+1))) = (1/2) * A002117 * A065465 = 0.529814898136... .

MATHEMATICA

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

PROG

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

CROSSREFS

Cf. A000203, A019554, A322483, A365346.

Cf. A002117, A065465.

KEYWORD

allocated

nonn,easy,mult

AUTHOR

Amiram Eldar, Sep 02 2023

STATUS

approved

editing

#1 by Amiram Eldar at Sat Sep 02 05:37:36 EDT 2023

NAME

allocated for Amiram Eldar

KEYWORD

allocated

STATUS

approved