A326040 -id:A326040 - cp's OEIS frontend

A326046 a(n) = gcd(n-A326039(n), A326040(n)-n).

1, 1, 1, 1, 4, 2, 3, 1, 1, 1, 1, 4, 12, 2, 1, 1, 8, 1, 3, 1, 5, 2, 1, 4, 1, 5, 1, 24, 28, 6, 15, 1, 1, 1, 1, 1, 36, 2, 1, 1, 40, 2, 3, 4, 4, 10, 1, 4, 1, 7, 15, 3, 4, 2, 19, 4, 1, 1, 1, 8, 60, 2, 1, 1, 1, 6, 3, 1, 1, 2, 35, 1, 72, 1, 1, 12, 1, 2, 3, 1, 1, 1, 1, 4, 1, 2, 1, 4, 8, 27, 5, 8, 29, 2, 7, 60, 48, 1, 1, 1, 100, 6, 3, 1, 1

Offset: 1

Antti Karttunen, Jun 06 2019

nonn

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

Cf. A000203, A008833, A326039, A326040, A326044, A326045.

Cf. also A009194, A325385, A325813, A325975, A326047, A326048, A326056, A326057, A326060, A326062, A326129, A326130, A326140, A326144.

A008833(n) = (n/core(n));
A326039(n) = A008833(sigma(n));
A326040(n) = (sigma(n)-A008833(sigma(n)));
A326046(n) = gcd(n-A326039(n), A326040(n)-n);

a(n) = gcd(A326044(n), A326045(n)) = gcd(n-A326039(n), A326040(n)-n).

A326039 Largest square dividing the sum of divisors of n: a(n) = A008833(sigma(n)).

Original entry on oeis.org

1, 1, 4, 1, 1, 4, 4, 1, 1, 9, 4, 4, 1, 4, 4, 1, 9, 1, 4, 1, 16, 36, 4, 4, 1, 1, 4, 4, 1, 36, 16, 9, 16, 9, 16, 1, 1, 4, 4, 9, 1, 16, 4, 4, 1, 36, 16, 4, 1, 1, 36, 49, 9, 4, 36, 4, 16, 9, 4, 4, 1, 16, 4, 1, 4, 144, 4, 9, 16, 144, 36, 1, 1, 1, 4, 4, 16, 4, 16, 1, 121, 9, 4, 16, 36, 4, 4, 36, 9, 9, 16, 4, 64, 144, 4, 36

Offset: 1

Antti Karttunen, Jun 05 2019

nonn

Antti Karttunen, Table of n, a(n) for n = 1..20000
Index entries for sequences related to sigma(n)

Cf. A000203, A008833, A326038, A326040.

Module[{sqs=Range[100,1,-1]^2},Table[SelectFirst[sqs,Divisible[ DivisorSigma[ 1,n],#]&],{n,100}]] (* Harvey P. Dale, Jul 29 2019 *)

A008833(n) = (n/core(n));
A326039(n) = A008833(sigma(n));

a(n) = A008833(A000203(n)) = A326038(n)^2.

a(n) = A000203(n) - A326040(n).

A326055 a(n) = n - {the largest square that divides n}.

Original entry on oeis.org

0, 1, 2, 0, 4, 5, 6, 4, 0, 9, 10, 8, 12, 13, 14, 0, 16, 9, 18, 16, 20, 21, 22, 20, 0, 25, 18, 24, 28, 29, 30, 16, 32, 33, 34, 0, 36, 37, 38, 36, 40, 41, 42, 40, 36, 45, 46, 32, 0, 25, 50, 48, 52, 45, 54, 52, 56, 57, 58, 56, 60, 61, 54, 0, 64, 65, 66, 64, 68, 69, 70, 36, 72, 73, 50, 72, 76, 77, 78, 64, 0, 81, 82, 80, 84

Offset: 1

Antti Karttunen, Jun 05 2019

nonn

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

Cf. A000203, A000290, A008833, A033879, A326040, A326054, A326056.

Table[n-Max[Select[Divisors[n],IntegerQ[Sqrt[#]]&]],{n,100}] (* Harvey P. Dale, Apr 21 2020 *)

A008833(n) = (n/core(n));
A326055(n) = (n-A008833(n));

a(n) = n - A008833(n).
a(n) = A326054(n) + A033879(n).
a(A000203(n)) = A326040(n).

A326045 a(n) is the sum of divisors of n, minus the largest square dividing that sum, minus n: a(n) = sigma(n) - A008833(sigma(n)) - n.

Original entry on oeis.org

-1, 0, -3, 2, 0, 2, -3, 6, 3, -1, -3, 12, 0, 6, 5, 14, -8, 20, -3, 21, -5, -22, -3, 32, 5, 15, 9, 24, 0, 6, -15, 22, -1, 11, -3, 54, 0, 18, 13, 41, 0, 38, -3, 36, 32, -10, -15, 72, 7, 42, -15, -3, -8, 62, -19, 60, 7, 23, -3, 104, 0, 18, 37, 62, 15, -66, -3, 49, 11, -70, -35, 122, 0, 39, 45, 60, 3, 86, -15, 105, -81, 35, -3

Offset: 1

Antti Karttunen, Jun 05 2019

sign

Antti Karttunen, Table of n, a(n) for n = 1..20000
Index entries for sequences related to sigma(n)

Cf. A000203, A008833, A326039, A326040, A326044, A326046.

A008833(n) = (n/core(n));
A326039(n) = A008833(sigma(n));
A326040(n) = (sigma(n)-A008833(sigma(n)));
A326045(n) = (A326040(n)-n);

a(n) = A326040(n) - n = sigma(n) - A008833(sigma(n)) - n.

A326038 Square root of the largest square dividing the sum of divisors of n: a(n) = A000188(sigma(n)).

Original entry on oeis.org

1, 1, 2, 1, 1, 2, 2, 1, 1, 3, 2, 2, 1, 2, 2, 1, 3, 1, 2, 1, 4, 6, 2, 2, 1, 1, 2, 2, 1, 6, 4, 3, 4, 3, 4, 1, 1, 2, 2, 3, 1, 4, 2, 2, 1, 6, 4, 2, 1, 1, 6, 7, 3, 2, 6, 2, 4, 3, 2, 2, 1, 4, 2, 1, 2, 12, 2, 3, 4, 12, 6, 1, 1, 1, 2, 2, 4, 2, 4, 1, 11, 3, 2, 4, 6, 2, 2, 6, 3, 3, 4, 2, 8, 12, 2, 6, 7, 3, 2, 1, 1, 6, 2, 1, 8

Offset: 1

Antti Karttunen, Jun 05 2019

nonn

Antti Karttunen, Table of n, a(n) for n = 1..65537
Index entries for sequences related to sigma(n)

Cf. A000188, A000196, A326038, A326039, A326040.

Table[Last[Select[Sqrt[#]&/@Divisors[DivisorSigma[1,n]],IntegerQ]],{n,120}] (* Harvey P. Dale, Jun 25 2022 *)

A000188(n) = core(n, 1)[2]; \\ From A000188
A326038(n) = A000188(sigma(n));

a(n) = A000188(A000203(n)) = A000196(A326039(n)).

cp's OEIS Frontend

A326046 a(n) = gcd(n-A326039(n), A326040(n)-n).

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A326039 Largest square dividing the sum of divisors of n: a(n) = A008833(sigma(n)).

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A326055 a(n) = n - {the largest square that divides n}.

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A326045 a(n) is the sum of divisors of n, minus the largest square dividing that sum, minus n: a(n) = sigma(n) - A008833(sigma(n)) - n.

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A326038 Square root of the largest square dividing the sum of divisors of n: a(n) = A000188(sigma(n)).

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Mathematica

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