A326126 -id:A326126 - cp's OEIS frontend

A326127 a(n) = A326126(n) - n, where A326126 gives the sum of all other divisors of n except the squarefree part of n.

-1, -1, -2, 2, -4, 0, -6, 5, 3, -2, -10, 13, -12, -4, -6, 14, -16, 19, -18, 17, -10, -8, -22, 30, 5, -10, 10, 21, -28, 12, -30, 29, -18, -14, -22, 54, -36, -16, -22, 40, -40, 12, -42, 29, 28, -20, -46, 73, 7, 41, -30, 33, -52, 60, -38, 50, -34, -26, -58, 93, -60, -28, 34, 62, -46, 12, -66, 41, -42, 4, -70, 121, -72, -34, 46, 45, -58, 12

Offset: 1

Antti Karttunen, Jun 09 2019

sign

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

Cf. A000203, A001065, A007913, A033879, A326126, A326128, A326129.

Cf. also A326143.

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 - n]; a[1] = -1; Array[a, 100] (* Amiram Eldar, Mar 21 2024 *)

PARI
```
A326127(n) = (sigma(n)-core(n)-n);
```

a(n) = A000203(n) - A007913(n) - n = A001065(n) - A007913(n).
a(n) = A326128(n) - A033879(n).
Sum_{k=1..n} a(k) ~ c * n^2, where c = Pi^2/20 - 1/2 = -0.00651977... . - Amiram Eldar, Mar 21 2024

A326129 a(n) = gcd(A326127(n), A326128(n)).

Original entry on oeis.org

1, 1, 2, 1, 4, 0, 6, 1, 1, 2, 10, 1, 12, 4, 6, 1, 16, 1, 18, 1, 10, 8, 22, 6, 1, 10, 2, 21, 28, 12, 30, 1, 18, 14, 22, 1, 36, 16, 22, 10, 40, 12, 42, 1, 4, 20, 46, 1, 1, 1, 30, 3, 52, 12, 38, 2, 34, 26, 58, 3, 60, 28, 2, 1, 46, 12, 66, 1, 42, 4, 70, 1, 72, 34, 2, 3, 58, 12, 78, 1, 1, 38, 82, 7, 62, 40, 54, 2, 88, 2, 70, 1, 58, 44, 70, 30

Offset: 1

Antti Karttunen, Jun 09 2019

nonn

Question: Are there any other numbers than those in A000396 that satisfy a(k) = A326128(k)?

See also comments in A336641, where all such k should reside. - Antti Karttunen, Jul 29 2020

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

Cf. A000396, A007913, A326126, A326127, A326128.

Cf. also A009194, A325385, A325813, A325975, A326046, A326047, A326048, A326056, A326057, A326060, A326062, A326130, A326140, A326144, A336641, A336645.

A326127(n) = (sigma(n)-core(n)-n);
A326128(n) = (n-core(n));
A326129(n) = gcd(A326127(n), A326128(n));

a(n) = n - A336645(n). - Antti Karttunen, Jul 29 2020

A326128 a(n) = n - A007913(n), where A007913 gives the squarefree part of n.

Original entry on oeis.org

0, 0, 0, 3, 0, 0, 0, 6, 8, 0, 0, 9, 0, 0, 0, 15, 0, 16, 0, 15, 0, 0, 0, 18, 24, 0, 24, 21, 0, 0, 0, 30, 0, 0, 0, 35, 0, 0, 0, 30, 0, 0, 0, 33, 40, 0, 0, 45, 48, 48, 0, 39, 0, 48, 0, 42, 0, 0, 0, 45, 0, 0, 56, 63, 0, 0, 0, 51, 0, 0, 0, 70, 0, 0, 72, 57, 0, 0, 0, 75, 80, 0, 0, 63, 0, 0, 0, 66, 0, 80, 0, 69, 0, 0, 0, 90, 0, 96, 88, 99, 0, 0, 0, 78, 0

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.

Cf. A007913, A033879, A326126, A326127, A326129, A336642.

Cf. also A066503.

f[p_, e_] := p^Mod[e, 2]; a[n_] := n - Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Mar 21 2024 *)

PARI
```
A326128(n) = (n-core(n));
```

a(n) = n - A007913(n).
a(n) = A326127(n) + A033879(n).
a(n) >= A066503(n).
a(n) = A007913(n) * A336642(n). - Antti Karttunen, Jul 28 2020
Sum_{k=1..n} a(k) ~ c * n^2, where c = 1/2 - Pi^2/30 = 0.171013... . - 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

A336645 a(n) = n - A326129(n).

Original entry on oeis.org

0, 1, 1, 3, 1, 6, 1, 7, 8, 8, 1, 11, 1, 10, 9, 15, 1, 17, 1, 19, 11, 14, 1, 18, 24, 16, 25, 7, 1, 18, 1, 31, 15, 20, 13, 35, 1, 22, 17, 30, 1, 30, 1, 43, 41, 26, 1, 47, 48, 49, 21, 49, 1, 42, 17, 54, 23, 32, 1, 57, 1, 34, 61, 63, 19, 54, 1, 67, 27, 66, 1, 71, 1, 40, 73, 73, 19, 66, 1, 79, 80, 44, 1, 77, 23, 46, 33, 86, 1, 88, 21

Offset: 1

Antti Karttunen, Jul 29 2020

nonn

It seems that A000396 gives all such n for which a(n) = A007913(n).

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, A000396, A007913, A326126, A326127, A326128, A326129.

A326127(n) = (sigma(n)-core(n)-n);
A326128(n) = (n-core(n));
A336645(n) = (n-gcd(A326127(n), A326128(n)));

a(n) = n - A326129(n) = n - gcd(A326127(n), A326128(n)).

A336641 Numbers k such that A007913(k) divides sigma(k) and A008833(k)-1 either divides A326127(k) (= sigma(k)-core(k)-k), or both are zero.

Original entry on oeis.org

6, 24, 28, 96, 120, 150, 294, 384, 496, 1014, 1536, 3276, 3750, 3780, 6144, 8128, 14406, 20328, 24576, 32760, 93750, 98304, 171366, 306180, 393216, 705894, 1241460, 1572864, 2343750, 6291456, 16380000, 24800580, 25165824, 28960854, 30387840, 33550336, 34588806, 58593750, 100663296, 165143160, 332226048, 402653184

Offset: 1

Antti Karttunen, Jul 28 2020

nonn

Numbers k such that A326128(k) = A326129(k) form a subsequence of this sequence. So far it is not known whether it contains any other terms apart from those of A000396. See comments in A326129.
Sequence is infinite because all numbers of the form 6*4^n (A002023) are present.
Question: Are there any odd terms?

Index entries for sequences where odd perfect numbers must occur, if they exist at all

Cf. A000203, A007913, A228058, A326126, A326127, A326128, A326129, A336642.
Cf. A000396, A002023 (subsequences).
Cf. also A336550 for a similar construction.

isA336641(n) = { my(c=core(n), s=sigma(n), u=((n/c)-1)); (!(s%c) && (gcd(u,(s-c-n))==u)); };

cp's OEIS Frontend

A326127 a(n) = A326126(n) - n, where A326126 gives the sum of all other divisors of n except the squarefree part of n.

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A326129 a(n) = gcd(A326127(n), A326128(n)).

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A326128 a(n) = n - A007913(n), where A007913 gives the squarefree part of 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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A336645 a(n) = n - A326129(n).

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A336641 Numbers k such that A007913(k) divides sigma(k) and A008833(k)-1 either divides A326127(k) (= sigma(k)-core(k)-k), or both are zero.

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