A326053 -id:A326053 - cp's OEIS frontend

A326054 a(n) = A326053(n) - n, where A326053 gives the sum of all other divisors of n except the largest square divisor.

-1, 0, 0, -1, 0, 5, 0, 3, -5, 7, 0, 12, 0, 9, 8, -1, 0, 12, 0, 18, 10, 13, 0, 32, -19, 15, 4, 24, 0, 41, 0, 15, 14, 19, 12, 19, 0, 21, 16, 46, 0, 53, 0, 36, 24, 25, 0, 60, -41, 18, 20, 42, 0, 57, 16, 60, 22, 31, 0, 104, 0, 33, 32, -1, 18, 77, 0, 54, 26, 73, 0, 87, 0, 39, 24, 60, 18, 89, 0, 90, -41, 43, 0, 136, 22, 45, 32, 88, 0, 135, 20

Offset: 1

Antti Karttunen, Jun 05 2019

sign

Antti Karttunen, Table of n, a(n) for n = 1..16384
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537
Index entries for sequences related to sigma(n)

Cf. A000203, A008833, A033879, A326053, A326055.

A008833(n) = (n/core(n));
A326053(n) = (sigma(n)-A008833(n));
A326054(n) = (A326053(n)-n);

a(n) = A326053(n) - n = (A000203(n) - A008833(n)) - n.

a(n) = A326055(n) - A033879(n).

A326056 a(n) = gcd(sigma(n)-A008833(n)-n, n-A008833(n)), where sigma is the sum of divisors of n, and A008833 is the largest square dividing n.

Original entry on oeis.org

1, 1, 2, 1, 4, 5, 6, 1, 5, 1, 10, 4, 12, 1, 2, 1, 16, 3, 18, 2, 10, 1, 22, 4, 19, 5, 2, 24, 28, 1, 30, 1, 2, 1, 2, 19, 36, 1, 2, 2, 40, 1, 42, 4, 12, 5, 46, 4, 41, 1, 10, 6, 52, 3, 2, 4, 2, 1, 58, 8, 60, 1, 2, 1, 2, 1, 66, 2, 2, 1, 70, 3, 72, 1, 2, 12, 2, 1, 78, 2, 41, 1, 82, 8, 2, 5, 2, 4, 88, 27, 10, 8, 2, 1, 2, 20, 96, 1, 6

Offset: 1

Antti Karttunen, Jun 05 2019

nonn

Composite numbers n such that a(n) = A326055(n) start as: 6, 28, 336, 496, 792, 8128, 31968, 3606912, ...

Nonsquare odd numbers n such that a(n) = abs(A326054(n)) start as: 21, 153, 301, 697, 1333, 1909, 1917, 2041, 3901, 4753, 24601, 24957, 26977, 29161, 29637, 56953, 67077, 96361, ...

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

Cf. A000203, A008833, A326053, A326054, A326055.

Cf. also A009194, A325385, A325813, A325975, A326046, A326047, A326048, A326057, A326060, A326062, A326129, A326130, A326140, A326144.

A008833(n) = (n/core(n));
A326053(n) = (sigma(n)-A008833(n));
A326054(n) = (A326053(n)-n);
A326055(n) = (n-A008833(n));
A326056(n) = gcd(A326054(n), A326055(n));

a(n) = gcd(A326054(n), A326055(n)) = gcd((A000203(n)-A008833(n))-n, n-A008833(n)).

A326126 Sum of all other divisors of n except the squarefree part of n: a(n) = sigma(n) - A007913(n).

Original entry on oeis.org

0, 1, 1, 6, 1, 6, 1, 13, 12, 8, 1, 25, 1, 10, 9, 30, 1, 37, 1, 37, 11, 14, 1, 54, 30, 16, 37, 49, 1, 42, 1, 61, 15, 20, 13, 90, 1, 22, 17, 80, 1, 54, 1, 73, 73, 26, 1, 121, 56, 91, 21, 85, 1, 114, 17, 106, 23, 32, 1, 153, 1, 34, 97, 126, 19, 78, 1, 109, 27, 74, 1, 193, 1, 40, 121, 121, 19, 90, 1, 181, 120, 44, 1, 203, 23, 46, 33, 158, 1, 224, 21

Offset: 1

Antti Karttunen, Jun 09 2019

nonn

Antti Karttunen, Table of n, a(n) for n = 1..16384
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537.
Index entries for sequences related to sigma(n).

Cf. A000203, A007913, A326127, A326128, A326129.

Cf. also A326053, A326061, A326142.

f1[p_, e_] := (p^(e + 1) - 1)/(p - 1); f2[p_, e_] := p^Mod[e, 2]; a[n_] := Module[{f = FactorInteger[n]}, Times @@ f1 @@@ f - Times @@ f2 @@@ f]; a[1] = 0; Array[a, 100] (* Amiram Eldar, Mar 21 2024 *)

PARI
```
A326126(n) = (sigma(n)-core(n));
```

a(n) = A000203(n) - A007913(n).

Sum_{k=1..n} a(k) ~ c * n^2, where c = Pi^2/20 = 0.4934802... . - Amiram Eldar, Mar 21 2024

A326142 Sum of all other divisors of n except its largest squarefree divisor: a(n) = sigma(n) - A007947(n).

Original entry on oeis.org

0, 1, 1, 5, 1, 6, 1, 13, 10, 8, 1, 22, 1, 10, 9, 29, 1, 33, 1, 32, 11, 14, 1, 54, 26, 16, 37, 42, 1, 42, 1, 61, 15, 20, 13, 85, 1, 22, 17, 80, 1, 54, 1, 62, 63, 26, 1, 118, 50, 83, 21, 72, 1, 114, 17, 106, 23, 32, 1, 138, 1, 34, 83, 125, 19, 78, 1, 92, 27, 74, 1, 189, 1, 40, 109, 102, 19, 90, 1, 176, 118, 44, 1, 182, 23, 46, 33

Offset: 1

Antti Karttunen, Jun 09 2019

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, A007947, A066503, A326143.
Cf. also A326053, A326061, A326126.
Cf. A013661, A065463.

rad[n_] := Times @@ FactorInteger[n][[;; , 1]]; a[n_] := DivisorSigma[1, n] - rad[n]; Array[a, 100] (* Amiram Eldar, Dec 05 2023 *)

A007947(n) = factorback(factorint(n)[, 1]);
A326142(n) = (sigma(n)-A007947(n));

a(n) = A000203(n) - A007947(n).
a(n) = n + A326143(n).
Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = A013661 - A065463 = 0.940491... . - Amiram Eldar, Dec 05 2023

A326061 Sum of all other divisors of n except the largest proper divisor. a(1) = 0 by convention.

Original entry on oeis.org

0, 2, 3, 5, 5, 9, 7, 11, 10, 13, 11, 22, 13, 17, 19, 23, 17, 30, 19, 32, 25, 25, 23, 48, 26, 29, 31, 42, 29, 57, 31, 47, 37, 37, 41, 73, 37, 41, 43, 70, 41, 75, 43, 62, 63, 49, 47, 100, 50, 68, 55, 72, 53, 93, 61, 92, 61, 61, 59, 138, 61, 65, 83, 95, 71, 111, 67, 92, 73, 109, 71, 159, 73, 77, 99, 102, 85, 129, 79, 146, 94, 85, 83

Offset: 1

Antti Karttunen, Jun 06 2019

nonn

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

Cf. A000203, A032742, A318505, A326062.

Cf. also A326053.

A032742(n) = if(1==n,n,n/vecmin(factor(n)[,1]));
A326061(n) = (sigma(n)-A032742(n));

a(n) = A000203(n) - A032742(n).

For n > 1, a(n) = n + A318505(n).

cp's OEIS Frontend

A326054 a(n) = A326053(n) - n, where A326053 gives the sum of all other divisors of n except the largest square divisor.

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A326056 a(n) = gcd(sigma(n)-A008833(n)-n, n-A008833(n)), where sigma is the sum of divisors of n, and A008833 is the largest square dividing n.

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A326126 Sum of all other divisors of n except the squarefree part of n: a(n) = sigma(n) - A007913(n).

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A326142 Sum of all other divisors of n except its largest squarefree divisor: a(n) = sigma(n) - A007947(n).

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A326061 Sum of all other divisors of n except the largest proper divisor. a(1) = 0 by convention.

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