A336642 -id:A336642 - cp's OEIS frontend

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

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

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)); };

A336644 a(n) = (n-rad(n)) / core(n), where rad(n) and core(n) give the squarefree kernel and squarefree part of n, respectively.

Original entry on oeis.org

0, 0, 0, 2, 0, 0, 0, 3, 6, 0, 0, 2, 0, 0, 0, 14, 0, 6, 0, 2, 0, 0, 0, 3, 20, 0, 8, 2, 0, 0, 0, 15, 0, 0, 0, 30, 0, 0, 0, 3, 0, 0, 0, 2, 6, 0, 0, 14, 42, 20, 0, 2, 0, 8, 0, 3, 0, 0, 0, 2, 0, 0, 6, 62, 0, 0, 0, 2, 0, 0, 0, 33, 0, 0, 20, 2, 0, 0, 0, 14, 78, 0, 0, 2, 0, 0, 0, 3, 0, 6, 0, 2, 0, 0, 0, 15, 0, 42, 6, 90, 0, 0, 0, 3, 0

Offset: 1

Antti Karttunen, Jul 28 2020

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, A007947, A008833, A066503, A336642, A336643.

A336644(n) = ((n-factorback(factorint(n)[, 1])) / core(n));

from math import prod
from sympy.ntheory.factor_ import primefactors, core
def A336644(n): return (n-prod(primefactors(n)))//core(n) # Chai Wah Wu, Dec 30 2021

a(n) = A066503(n) / A007913(n) = (n-A007947(n)) / A007913(n).

a(n) = A008833(n) - A336643(n).

cp's OEIS Frontend

A326128 a(n) = n - A007913(n), where A007913 gives the squarefree part of 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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A336644 a(n) = (n-rad(n)) / core(n), where rad(n) and core(n) give the squarefree kernel and squarefree part of n, respectively.

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