A335341 -id:A335341 - cp's OEIS frontend

A336563 Sum of proper divisors of n that are divisible by every prime that divides n.

0, 0, 0, 2, 0, 0, 0, 6, 3, 0, 0, 6, 0, 0, 0, 14, 0, 6, 0, 10, 0, 0, 0, 18, 5, 0, 12, 14, 0, 0, 0, 30, 0, 0, 0, 36, 0, 0, 0, 30, 0, 0, 0, 22, 15, 0, 0, 42, 7, 10, 0, 26, 0, 24, 0, 42, 0, 0, 0, 30, 0, 0, 21, 62, 0, 0, 0, 34, 0, 0, 0, 96, 0, 0, 15, 38, 0, 0, 0, 70, 39, 0, 0, 42, 0, 0, 0, 66, 0, 30, 0, 46, 0, 0, 0, 90

Offset: 1

Antti Karttunen, Jul 27 2020

nonn

Antti Karttunen, Table of n, a(n) for n = 1..10800
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537

Cf. A001065, A003557, A007947, A057723, A033879, A065487, A335341, A336564, A336565, A336566, A336567.

Cf. A005117 (positions of zeros).

f[p_, e_] := (p^(e + 1) - 1)/(p - 1) - 1; a[1] = 0; a[n_] := Times @@ f @@@ FactorInteger[n] - n; Array[a, 100] (* Amiram Eldar, May 06 2023 *)

A007947(n) = factorback(factorint(n)[, 1]);
A057723(n) = { my(r=A007947(n)); (r*sigma(n/r)); };
A336563(n) = (A057723(n)-n);
\\ Or just as:
A336563(n) = { my(x=A007947(n),y = n/x); (x*(sigma(y)-y)); };

a(n) = A057723(n) - n.
a(n) = A007947(n) * A336567(n) = A007947(n) * A001065(A003557(n)).
a(n) = A336564(n) - A033879(n).
Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = A065487 - 1 = 0.231291... . - Amiram Eldar, Dec 07 2023

A336649 Sum of divisors of A336651(n) (odd part of n divided by its largest squarefree divisor).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 6, 1, 13, 1, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 8, 6, 1, 1, 1, 13, 1, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 6, 1, 1, 1, 1, 1, 40, 1, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 8, 4, 6, 1, 1, 1, 1, 1

Offset: 1

Antti Karttunen, Jul 30 2020

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A000203, A000265, A204455, A335341, A336651, A336652.

f[2, e_] := 1; f[p_, e_] := (p^e - 1)/(p-1); a[1] = 1; a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100] (* Amiram Eldar, Dec 07 2020 *)

A336649(n) = { my(f=factor(n)); prod(i=1, #f~, if((2==f[i,1])||(1==f[i,2]),1,(((f[i,1]^(f[i,2]))-1) / (f[i,1]-1)))); };

A000265(n) = (n>>valuation(n,2));
A335341(n) = if(1==n,n,sigma(n/factorback(factorint(n)[, 1])));
A336649(n) = A335341(A000265(n));

Multiplicative with a(2^e) = 1, a(p^1) = 1 and a(p^e) = (p^e - 1)/(p-1) if e > 1.
a(n) = A000203(A336651(n)) = A335341(A000265(n)).
a(n) = A336652(n) / A204455(n).
Dirichlet g.f.: zeta(s-1) * zeta(s) * (1 - 1/(1-2^s+2^(2*s-1))) * Product_{p prime} (1 - 1/p^(s-1) + 1/p^(2*s-1)). - Amiram Eldar, Dec 18 2023

A336567 Sum of proper divisors of {n divided by its largest squarefree divisor}.

Original entry on oeis.org

0, 0, 0, 1, 0, 0, 0, 3, 1, 0, 0, 1, 0, 0, 0, 7, 0, 1, 0, 1, 0, 0, 0, 3, 1, 0, 4, 1, 0, 0, 0, 15, 0, 0, 0, 6, 0, 0, 0, 3, 0, 0, 0, 1, 1, 0, 0, 7, 1, 1, 0, 1, 0, 4, 0, 3, 0, 0, 0, 1, 0, 0, 1, 31, 0, 0, 0, 1, 0, 0, 0, 16, 0, 0, 1, 1, 0, 0, 0, 7, 13, 0, 0, 1, 0, 0, 0, 3, 0, 1, 0, 1, 0, 0, 0, 15, 0, 1, 1, 8, 0, 0, 0, 3, 0

Offset: 1

Antti Karttunen, Jul 27 2020

nonn

Antti Karttunen, Table of n, a(n) for n = 1..16384
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537

Cf. A000203, A001065, A003557, A007947, A057723, A335341, A336563.

A007947(n) = factorback(factorint(n)[, 1]);
A057723(n) = { my(r=A007947(n)); (r*sigma(n/r)); };
A336563(n) = (A057723(n)-n);
A336567(n) = (A336563(n)/A007947(n));

a(n) = A001065(A003557(n)).

a(n) = A335341(n) - A003557(n) = A336563(n) / A007947(n).

A376216 Numbers whose sum of powerful divisors (including 1) is even.

Original entry on oeis.org

9, 18, 25, 36, 45, 49, 50, 63, 72, 75, 81, 90, 98, 99, 100, 117, 121, 126, 144, 147, 150, 153, 162, 169, 171, 175, 180, 196, 198, 200, 207, 225, 234, 242, 245, 252, 261, 275, 279, 288, 289, 294, 300, 306, 315, 324, 325, 333, 338, 342, 350, 360, 361, 363, 369, 387, 392, 396, 400

Offset: 1

Amiram Eldar, Sep 15 2024

The sequence of numbers whose number of powerful divisors (including 1, A005361) is even is A072587, which is the sequence of numbers that are not exponentially odd (A268335).
The primitive terms of this sequence are the powerful terms (A376217). If m is a powerful term then k*m is a term of this sequence for all squarefree numbers k that are coprime to m.
Numbers that have at least one odd prime factor in their prime factorization that has an even exponent.
Numbers whose odd part (A000265) is not an exponentially odd number (A268335).
Also, numbers k such that A335341(k) is even.
The asymptotic density of this sequence is 1 - (6/5) * A065463 = 0.15466935880100128871... .

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

Subsequence of A013929.

Cf. A000265, A005117, A005361, A065463, A072587, A183097, A268335, A335341, A376217 (subsequence).

q[n_] := Module[{f = Select[FactorInteger[n], First[#] == 2 || Last[#] > 1 &], i = 2 - Mod[n, 2]}, Length[f] > 0 && AnyTrue[f[[i;;-1, 2]], EvenQ]]; Select[Range[400], q]

is(k) = {my(f = factor(k), i = 1 + !(k % 2)); #select(x -> !(x%2), f[i..#f~,2]) > 0;}

A383717 Dirichlet g.f.: Product_{p prime} (1 + 1/p^(s-1) + 1/p^(2*s-1)).

Original entry on oeis.org

1, 2, 3, 2, 5, 6, 7, 0, 3, 10, 11, 6, 13, 14, 15, 0, 17, 6, 19, 10, 21, 22, 23, 0, 5, 26, 0, 14, 29, 30, 31, 0, 33, 34, 35, 6, 37, 38, 39, 0, 41, 42, 43, 22, 15, 46, 47, 0, 7, 10, 51, 26, 53, 0, 55, 0, 57, 58, 59, 30, 61, 62, 21, 0, 65, 66, 67, 34, 69, 70, 71, 0, 73

Offset: 1

Vaclav Kotesovec, May 07 2025

Vaclav Kotesovec, Table of n, a(n) for n = 1..10000

Cf. A007947, A056552, A335341, A336649.

f[p_, e_] := If[e < 3, p, 0]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, May 07 2025 *)

for(n=1, 100, print1(direuler(p=2, n, (1 + X*p + X^2*p))[n], ", "))

Sum_{k=1..n} a(k) ~ c * n^2/2, where c = Product_{p prime} (1 - 1/p^2 + 1/p^3 - 1/p^4) = 0.684286924186862318141968725791218083472312736723163777284618226290055...

Multiplicative with a(p^e) = p is e <= 2, and 0 otherwise. - Amiram Eldar, May 07 2025

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A336563 Sum of proper divisors of n that are divisible by every prime that divides n.

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A336649 Sum of divisors of A336651(n) (odd part of n divided by its largest squarefree divisor).

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A336567 Sum of proper divisors of {n divided by its largest squarefree divisor}.

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A376216 Numbers whose sum of powerful divisors (including 1) is even.

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A383717 Dirichlet g.f.: Product_{p prime} (1 + 1/p^(s-1) + 1/p^(2*s-1)).

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