A326142 -id:A326142 - cp's OEIS frontend

A326143 a(n) = A326142(n) - n, where A326142 gives the sum of all other divisors of n except its largest squarefree divisor.

-1, -1, -2, 1, -4, 0, -6, 5, 1, -2, -10, 10, -12, -4, -6, 13, -16, 15, -18, 12, -10, -8, -22, 30, 1, -10, 10, 14, -28, 12, -30, 29, -18, -14, -22, 49, -36, -16, -22, 40, -40, 12, -42, 18, 18, -20, -46, 70, 1, 33, -30, 20, -52, 60, -38, 50, -34, -26, -58, 78, -60, -28, 20, 61, -46, 12, -66, 24, -42, 4, -70, 117, -72, -34, 34, 26, -58, 12

Offset: 1

Antti Karttunen, Jun 09 2019

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

Cf. A000203, A001065, A007947, A033879, A066503, A326142, A326144.
Cf. also A326054, A326127.
Cf. A013661, A065463.

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

A007947(n) = factorback(factorint(n)[, 1]);
A326143(n) = (sigma(n)-A007947(n)-n);

a(n) = A326142(n) - n = (A000203(n)-A007947(n)) - n = A001065(n) - A007947(n).
a(n) = A066503(n) - A033879(n).
Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = A013661 - A065463 - 1 = -0.0595081... . - Amiram Eldar, Dec 05 2023

A326144 a(n) = gcd(A066503(n), A326143(n)) = gcd(n - A007947(n), sigma(n) - A007947(n) - n).

Original entry on oeis.org

1, 1, 2, 1, 4, 0, 6, 1, 1, 2, 10, 2, 12, 4, 6, 1, 16, 3, 18, 2, 10, 8, 22, 6, 1, 10, 2, 14, 28, 12, 30, 1, 18, 14, 22, 1, 36, 16, 22, 10, 40, 12, 42, 2, 6, 20, 46, 14, 1, 1, 30, 2, 52, 12, 38, 2, 34, 26, 58, 6, 60, 28, 2, 1, 46, 12, 66, 2, 42, 4, 70, 3, 72, 34, 2, 2, 58, 12, 78, 2, 1, 38, 82, 14, 62, 40, 54, 2, 88, 6, 70, 2

Offset: 1

Antti Karttunen, Jun 09 2019

nonn

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

Cf. A000203, A007947, A066503, A326142, A326143, A326145.

Cf. also A009194, A325385, A325813, A325975, A326046, A326047, A326048, A326056, A326057, A326060, A326062, A326129, A326130, A326140.

rad[n_] := Times @@ (First /@ FactorInteger@n);
a[n_] := GCD[n - rad[n], DivisorSigma[1, n] - rad[n] - n];
Array[a, 100] (* Jean-François Alcover, Dec 01 2021 *)

A007947(n) = factorback(factorint(n)[, 1]);
A066503(n) = (n - A007947(n));
A326143(n) = (sigma(n)-A007947(n)-n);
A326144(n) = gcd(A066503(n), A326143(n));

a(n) = gcd(A066503(n), A326143(n)) = gcd(n-A007947(n), A000203(n)-A007947(n)-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

A326145 Numbers n for which n - A007947(n) is equal to gcd(n - A007947(n), sigma(n) - A007947(n) - n).

Original entry on oeis.org

6, 28, 496, 936, 1638, 8128, 33550336

Offset: 1

Antti Karttunen, Jun 09 2019

Numbers n such that either A066503(n) and A326143(n) are both zero or A066503(n) is not zero and divides A326143(n).
Question: Are there any odd terms?
No other terms < 2^31.

Index entries for sequences where any odd perfect numbers must occur

Cf. A000203, A000396 (a subsequence), A007947, A066503, A228058, A326142, A326143, A326144.

A007947(n) = factorback(factorint(n)[, 1]);
isA326145(n) = { my(b=A007947(n), t=n-b, u = (sigma(n)-b)-n); (gcd(t,u)==t); };
\\ Or alternatively as:
isA326145(n) = { my(t=A326143(n), u=A066503(n)); ((!u && !t)||(u && !(t%u))); };

A338790 a(n) = rad(n)^2 - sigma(n), where rad(n) is the squarefree kernel of n (A007947) and sigma(n) is the sum of divisors of n (A000203).

Original entry on oeis.org

0, 1, 5, -3, 19, 24, 41, -11, -4, 82, 109, 8, 155, 172, 201, -27, 271, -3, 341, 58, 409, 448, 505, -24, -6, 634, -31, 140, 811, 828, 929, -59, 1041, 1102, 1177, -55, 1331, 1384, 1465, 10, 1639, 1668, 1805, 400, 147, 2044, 2161, -88, -8, 7, 2529, 578, 2755, -84, 2953

Offset: 1

Michel Marcus, Nov 09 2020

sign

It is conjectured that only 1 and 1782 satisfy a(x) = 0.

R. K. Guy, Unsolved Problems in Theory of Numbers, Springer-Verlag, Third Edition, 2004, B11.

Michel Marcus, Table of n, a(n) for n = 1..10000
K. Broughan, J.-M. De Koninck, I. Kátai, F. Luca, On integers for which the sum of divisors is the square of the squarefree core, J. Integer Seq., 15 (2012), pp. 1-12.
Yong-Gao Chen, and Xin Tong, On a conjecture of de Koninck, Journal of Number Theory, Volume 154, September 2015, Pages 324-364. Beware of typo 1728.

Cf. A000203, A007947, A078615, A326142.

a:= n-> mul(i[1], i=ifactors(n)[2])^2-numtheory[sigma](n):
seq(a(n), n=1..60);  # Alois P. Heinz, Nov 09 2020

a(n) = my(f=factor(n)); factorback(f[, 1])^2 - sigma(f);

a(n) = A007947(n)^2 - A000203(n).

a(n) = A078615(n) - A000203(n).

cp's OEIS Frontend

A326143 a(n) = A326142(n) - n, where A326142 gives the sum of all other divisors of n except its largest squarefree divisor.

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A326144 a(n) = gcd(A066503(n), A326143(n)) = gcd(n - A007947(n), sigma(n) - A007947(n) - 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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A326145 Numbers n for which n - A007947(n) is equal to gcd(n - A007947(n), sigma(n) - A007947(n) - n).

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A338790 a(n) = rad(n)^2 - sigma(n), where rad(n) is the squarefree kernel of n (A007947) and sigma(n) is the sum of divisors of n (A000203).

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