A284118 -id:A284118 - cp's OEIS frontend

A318674 Sum of squarefree divisors of n that have an even number of prime factors.

1, 1, 1, 1, 1, 7, 1, 1, 1, 11, 1, 7, 1, 15, 16, 1, 1, 7, 1, 11, 22, 23, 1, 7, 1, 27, 1, 15, 1, 32, 1, 1, 34, 35, 36, 7, 1, 39, 40, 11, 1, 42, 1, 23, 16, 47, 1, 7, 1, 11, 52, 27, 1, 7, 56, 15, 58, 59, 1, 32, 1, 63, 22, 1, 66, 62, 1, 35, 70, 60, 1, 7, 1, 75, 16, 39, 78, 72, 1, 11, 1, 83, 1, 42, 86, 87, 88, 23, 1, 32, 92, 47

Offset: 1

Antti Karttunen, Sep 04 2018

nonn

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

Cf. A023900, A048250, A166142, A284118, A291784, A318675, A318676.

a[n_] := DivisorSum[n, # &, MoebiusMu[#] == 1 &]; Array[a, 100] (* Amiram Eldar, Jun 06 2025 *)

A318674(n) = sumdiv(n,d,(1==moebius(d))*d);

a(n) = Sum_{d|n} [A008683(d) > 0]*d.
a(n) = A048250(n) - A318675(n).
For all n >= 1, a(n) <= A318676(n).
a(n) = (A048250(n) + A023900(n))/2. - Amiram Eldar, Jun 06 2025

A290479 Product of nonprime squarefree divisors of n.

Original entry on oeis.org

1, 1, 1, 1, 1, 6, 1, 1, 1, 10, 1, 6, 1, 14, 15, 1, 1, 6, 1, 10, 21, 22, 1, 6, 1, 26, 1, 14, 1, 27000, 1, 1, 33, 34, 35, 6, 1, 38, 39, 10, 1, 74088, 1, 22, 15, 46, 1, 6, 1, 10, 51, 26, 1, 6, 55, 14, 57, 58, 1, 27000, 1, 62, 21, 1, 65, 287496, 1, 34, 69, 343000, 1, 6, 1, 74, 15, 38, 77, 474552, 1, 10

Offset: 1

Ilya Gutkovskiy, Aug 03 2017

nonn

			a(30) = 27000 because 30 has 8 divisors {1, 2, 3, 5, 6, 10, 15, 30} among which 5 are nonprime squarefree {1, 6, 10, 15, 30} and 1*6*10*15*30 = 27000.

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

Cf. A000005, A000469, A006881 (fixed points), A007947, A007955, A007956, A061537, A062758, A078599, A087652, A126192, A136655, A183091, A284118.

Table[Product[d, {d, Select[Divisors[n], !PrimeQ[#] && SquareFreeQ[#] &]}], {n, 80}]
Table[Last[Select[Divisors[n], SquareFreeQ]]^(DivisorSigma[0, Last[Select[Divisors[n], SquareFreeQ]]]/2 - 1), {n, 80}]

A290479(n) = if(1==n, n, my(r=factorback(factorint(n)[, 1])); (r^((numdiv(r)/2)-1))); \\ Antti Karttunen, Aug 06 2018

a(n) = A078599(n)/A007947(n).
a(n) = rad(n)^(d(rad(n))/2-1), where d() is the number of divisors of n (A000005) and rad() is the squarefree kernel of n (A007947).
a(n) = 1 if n is a prime power.

A358077 Sum of the nonprime divisors of n whose divisor complement is squarefree.

Original entry on oeis.org

1, 1, 1, 4, 1, 7, 1, 12, 9, 11, 1, 22, 1, 15, 16, 24, 1, 33, 1, 34, 22, 23, 1, 48, 25, 27, 36, 46, 1, 62, 1, 48, 34, 35, 36, 72, 1, 39, 40, 72, 1, 84, 1, 70, 69, 47, 1, 96, 49, 85, 52, 82, 1, 108, 56, 96, 58, 59, 1, 142, 1, 63, 93, 96, 66, 128, 1, 106, 70, 130, 1, 144, 1, 75

Offset: 1

Wesley Ivan Hurt, Oct 29 2022

nonn

			a(8) = 12. The nonprime divisors of 8 whose divisor complements are squarefree are 4 and 8 and their sum is 12.

Antti Karttunen, Table of n, a(n) for n = 1..20000

Cf. A005117, A284118.

a[n_] := DivisorSum[n, # &, ! PrimeQ[#] && SquareFreeQ[n/#] &]; Array[a, 100] (* Amiram Eldar, Oct 30 2022 *)

a(n) = sumdiv(n, d, if (!isprime(d) && issquarefree(n/d), d)); \\ Michel Marcus, Oct 30 2022

a(n) = Sum_{d|n, nonprime d, n/d squarefree} d.

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A318674 Sum of squarefree divisors of n that have an even number of prime factors.

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A290479 Product of nonprime squarefree divisors of n.

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A358077 Sum of the nonprime divisors of n whose divisor complement is squarefree.

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