A008833 -id:A008833 - cp's OEIS frontend

A056192 a(n) = n divided by its characteristic cube divisor A056191.

1, 2, 3, 4, 5, 6, 7, 1, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 3, 25, 26, 1, 28, 29, 30, 31, 4, 33, 34, 35, 36, 37, 38, 39, 5, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 2, 55, 7, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 9, 73, 74, 75

Offset: 1

Labos Elemer, Aug 02 2000

Different from A056552: e.g. a(16) = 16, while A056552(16) = 2.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000188, A008833, A007913, A055229, A055231, A056552, A056191.

f[p_, e_] := If[EvenQ[e], p^e, If[e == 1, p, p^(e - 3)]]; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Sep 06 2020 *)

a(n) = n/A055229(n)^3 = n/g^3=n/ggg and n=(LL)*(ggg)*f=L^2*g^3*f=LL*a(n)^3*f, so n=L^2*(g*3)*f, where L=A000188(n)/A055229(n), f=A055231(n), g=A055231(n).
Multiplicative with a(p^e)=p^e for even e, a(p)=p, a(p^e)=p^(e-3) for odd e>1. - Vladeta Jovovic, Apr 30 2002
Sum_{k=1..n} a(k) ~ c * n^2, where c = (Pi^2/12) * Product_{p prime} (1 - 1/p^2 - 1/p^3 + 1/p^4 + 1/p^6 - 1/p^7) = 0.4462648652... . - Amiram Eldar, Nov 13 2022

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

Original entry on oeis.org

-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).

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).

A334109 a(n) = A329697(A225546(n)).

Original entry on oeis.org

0, 0, 0, 1, 0, 0, 0, 1, 2, 0, 0, 1, 0, 0, 0, 1, 0, 2, 0, 1, 0, 0, 0, 1, 4, 0, 2, 1, 0, 0, 0, 1, 0, 0, 0, 3, 0, 0, 0, 1, 0, 0, 0, 1, 2, 0, 0, 1, 8, 4, 0, 1, 0, 2, 0, 1, 0, 0, 0, 1, 0, 0, 2, 2, 0, 0, 0, 1, 0, 0, 0, 3, 0, 0, 4, 1, 0, 0, 0, 1, 2, 0, 0, 1, 0, 0, 0, 1, 0, 2, 0, 1, 0, 0, 0, 1, 0, 8, 2, 5, 0, 0, 0, 1, 0

Offset: 1

Antti Karttunen, Apr 29 2020

nonn

Conjecture: Each k >= 0 occurs for the first time at A334110(k) = A019565(k)^2. Note that each k must occur first time on square n, because of the identity a(n) = a(A008833(n)). However, is there any reason to exclude squares with prime exponents > 2 from the candidates? See also comments in A334204.

Antti Karttunen, Table of n, a(n) for n = 1..10200
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537

Cf. A001248, A005117 (positions of zeros), A334110.
Cf. A003961, A008833, A019565, A059895, A225546, A329697, A331590, A331736, A334204, A334860.
Cf. also A334107, A334108.

Map[-1 + Length@ NestWhileList[# - #/FactorInteger[#][[-1, 1]] &, #, # != 2^IntegerExponent[#, 2] &] &, Array[If[# == 1, 1, Times @@ Flatten@ Map[Function[{p, e}, Map[Prime[Log2@ # + 1]^(2^(PrimePi@ p - 1)) &, DeleteCases[NumberExpand[e, 2], 0]]] @@ # &, FactorInteger[#]]] &, 105] ] (* Michael De Vlieger, May 26 2020 *)

A019565(n) = factorback(vecextract(primes(logint(n+!n, 2)+1), n));
A329697(n) = if(!bitand(n,n-1),0,1+A329697(n-(n/vecmax(factor(n)[, 1]))));
A334109(n) = { my(f=factor(n),pis=apply(primepi,f[,1]),es=f[,2]); sum(k=1,#f~,(2^(pis[k]-1))*A329697(A019565(es[k]))); };

Additive with a(prime(i)^j) = A000079(i-1) * A329697(A019565(j)), a(m*n) = a(m)+a(n) if gcd(m,n) = 1.
Alternatively, additive with a(prime(i)^(2^k)) = 2^(i-1) * A329697(prime(k+1)), a(m*n) = a(m)+a(n) if A059895(m,n) = 1. - Peter Munn, May 04 2020
a(n) = A329697(A225546(n)) = A329697(A331736(n)).
a(n) = a(A008833(n)).
For all n >= 0, a(A334110(n)) = n, a(A334860(n)) = A334204(n).
a(A331590(m,k)) = a(m) + a(k); a(A003961(n)) = 2*a(n). - Peter Munn, Apr 30 2020

A334871 Number of steps needed to reach 1 when starting from n and iterating with A334870.

Original entry on oeis.org

0, 1, 2, 2, 4, 3, 8, 3, 3, 5, 16, 4, 32, 9, 6, 3, 64, 4, 128, 6, 10, 17, 256, 5, 5, 33, 5, 10, 512, 7, 1024, 4, 18, 65, 12, 4, 2048, 129, 34, 7, 4096, 11, 8192, 18, 7, 257, 16384, 5, 9, 6, 66, 34, 32768, 6, 20, 11, 130, 513, 65536, 8, 131072, 1025, 11, 4, 36, 19, 262144, 66, 258, 13, 524288, 5, 1048576, 2049, 7, 130

Offset: 1

Antti Karttunen, Jun 08 2020

nonn

Distance of n from the root (1) in binary trees like A334860 and A334866.
Each n > 0 occurs 2^(n-1) times.
a(n) is the size of the inner lining of the integer partition with Heinz number A225546(n), which is also the size of the largest hook of the same partition. (After Gus Wiseman's Apr 02 2019 comment in A252464).

Antti Karttunen, Table of n, a(n) for n = 1..10000

Cf. A007913, A008833, A070939, A225546, A252464, A299090, A334859, A334860, A334865, A334866, A334869, A334870, A334872.

A334870(n) = if(issquare(n),sqrtint(n),my(c=core(n), m=n); forprime(p=2, , if(!(c % p), m/=p; break, m*=p)); (m));
A334871(n) = { my(s=0); while(n>1,s++; n = A334870(n)); (s); };

\\ Much faster:
A048675(n) = { my(f = factor(n)); sum(k=1, #f~, f[k, 2]*2^primepi(f[k, 1]))/2; };
A334871(n) = { my(s=0); while(n>1, if(issquare(n), s++; n = sqrtint(n), s += A048675(core(n)); n /= core(n))); (s); };

a(1) = 0; for n > 1, a(n) = 1 + a(A334870(n)).
a(n) = A252464(A225546(n)).
a(n) = A048675(A007913(n)) + a(A008833(n)).
For n > 1, a(n) = 1 + A048675(A007913(n)) + a(A000188(n)).
For n > 1, a(n) = A070939(A334859(n)) = A070939(A334865(n)).
For all n >= 1, a(n) >= A299090(n).
For all n >= 1, a(n) >= A334872(n).

A335324 Square part of 4th-power-free part of n.

Original entry on oeis.org

1, 1, 1, 4, 1, 1, 1, 4, 9, 1, 1, 4, 1, 1, 1, 1, 1, 9, 1, 4, 1, 1, 1, 4, 25, 1, 9, 4, 1, 1, 1, 1, 1, 1, 1, 36, 1, 1, 1, 4, 1, 1, 1, 4, 9, 1, 1, 1, 49, 25, 1, 4, 1, 9, 1, 4, 1, 1, 1, 4, 1, 1, 9, 4, 1, 1, 1, 4, 1, 1, 1, 36, 1, 1, 25, 4, 1, 1, 1, 1, 1, 1, 1, 4, 1

Offset: 1

Peter Munn, May 31 2020

Equivalently, biquadratefree (4th-power-free) part of square part of n.
Multiplicative. The terms are squares of squarefree numbers (A062503).
Every positive integer n is the product of a unique subset S_n of the terms of A050376 (sometimes called Fermi-Dirac primes). a(n) is the product of the members of S_n that are squares of prime numbers (A001248).

			Removing the 4th powers from 192 = 2^6 * 3^1 gives 2^(6 - 4) * 3^1 = 2^2 * 3 = 12. So the 4th-power-free part of 192 is 12. The square part of 12 (largest square dividing 12) is 4. So a(192) = 4.

Michel Marcus, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Biquadratefree.
Eric Weisstein's World of Mathematics, Square part.

A007913, A008833, A008835, A053165 are used in formulas defining the sequence.
Column 1 of A352780.
Range of values is A062503.
Positions of 1's: A252895.
Related to A038500 by A225546.
The formula section details how the sequence maps the terms of A003961, A331590.
Cf. A001248, A050376, A083730.
Cf. A002117, A013662, A078434.

f[p_, e_] := p^(2*Floor[e/2] - 4*Floor[e/4]); a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100] (* Amiram Eldar, Jun 01 2020 *)

A053165(n)=my(f=factor(n)); f[, 2]=f[, 2]%4; factorback(f);
a(n) = my(m=A053165(n)); m/core(m); \\ Michel Marcus, Jun 01 2020

from math import prod
from sympy import factorint
def A335324(n): return prod(p**(e&2) for p, e in factorint(n).items()) # Chai Wah Wu, Aug 07 2024

a(n) = A053165(A008833(n)) = A008833(A053165(n)).
a(n) = A053165(n) / A007913(n).
a(n) = A008833(n) / A008835(n).
n = A007913(n) * a(n) * A008835(n).
a(n) = A225546(A038500(A225546(n))).
a(n^2) = A007913(n)^2.
a(A003961(n)) = A003961(a(n)).
a(A331590(n, k)) = A331590(a(n), a(k)).
a(p^e) = p^(2*floor(e/2) - 4*floor(e/4)). - Amiram Eldar, Jun 01 2020
From Amiram Eldar, Sep 21 2023: (Start)
Dirichlet g.f.: zeta(s) * zeta(2*s-2) * zeta(4*s)/(zeta(2*s) * zeta(4*s-4)).
Sum_{k=1..n} a(k) ~ (4*zeta(3/2)*zeta(4))/(21*zeta(3)) * n^(3/2). (End)

A366245 The largest infinitary divisor of n that is a term of A366243.

Original entry on oeis.org

1, 1, 1, 4, 1, 1, 1, 4, 9, 1, 1, 4, 1, 1, 1, 1, 1, 9, 1, 4, 1, 1, 1, 4, 25, 1, 9, 4, 1, 1, 1, 1, 1, 1, 1, 36, 1, 1, 1, 4, 1, 1, 1, 4, 9, 1, 1, 1, 49, 25, 1, 4, 1, 9, 1, 4, 1, 1, 1, 4, 1, 1, 9, 4, 1, 1, 1, 4, 1, 1, 1, 36, 1, 1, 25, 4, 1, 1, 1, 1, 1, 1, 1, 4, 1

Offset: 1

Amiram Eldar, Oct 05 2023

First differs from A335324 at n = 256.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A063695, A335324, A366242, A366243, A366244.

See the formula section for the relationships with A008833, A046100, A059895, A059896, A059897, A225546, A248101, A352780.

f[p_, e_] := p^BitAnd[e, Sum[2^k, {k, 1, Floor@ Log2[e], 2}]]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

s(e) = -sum(k = 1, e, (-2)^k*floor(e/2^k));
a(n) = {my(f = factor(n)); prod(i = 1, #f~, f[i,1]^s(f[i,2]));}

Multiplicative with a(p^e) = p^A063695(e).
a(n) = n / A366244(n).
a(n) >= 1, with equality if and only if n is a term of A366242.
a(n) <= n, with equality if and only if n is a term of A366243.
From Peter Munn, Jan 09 2025: (Start)
a(n) = max({k in A366243 : A059895(k, n) = k}).
a(n) = Product_{k >= 0} A352780(n, 2k+1).
Also defined by:
- for n in A046100, a(n) = A008833(n);
- a(n^4) = (a(n))^4;
- a(A059896(n,k)) = A059896(a(n), a(k)).
Other identities:
a(n) = sqrt(A366244(n^2)).
a(A059897(n,k)) = A059897(a(n), a(k)).
a(A225546(n)) = A225546(A248101(n)).
(End)

A068976 a(n) = Sum_{d | n} d/core(d) where core(x) is the smallest number such that x*core(x) is a square.

Original entry on oeis.org

1, 2, 2, 6, 2, 4, 2, 10, 11, 4, 2, 12, 2, 4, 4, 26, 2, 22, 2, 12, 4, 4, 2, 20, 27, 4, 20, 12, 2, 8, 2, 42, 4, 4, 4, 66, 2, 4, 4, 20, 2, 8, 2, 12, 22, 4, 2, 52, 51, 54, 4, 12, 2, 40, 4, 20, 4, 4, 2, 24, 2, 4, 22, 106, 4, 8, 2, 12, 4, 8, 2, 110, 2, 4, 54, 12, 4, 8, 2, 52, 101, 4, 2, 24, 4, 4, 4

Offset: 1

Benoit Cloitre, Apr 06 2002

More generally, a(n,m) = Sum_{d divides n} gcd(d,n/d)^m is multiplicative with a(p^e,m) = (p^(m*e/2)*(p^m+1)-2)/(p^m-1) if e is even else 2*(p^(m*(e+1)/2)-1)/(p^m-1). - Vladeta Jovovic, May 30 2003

Robert Israel, Table of n, a(n) for n = 1..10000
Vaclav Kotesovec, Graph - the asymptotic ratio
R. Sivaramakrishnan, A. Somayajulu, H. Scheid and E. A. Bender, A Number-Theoretic Identity (Advanced Problem 5446 and solutions), American Mathematical Monthly 74 (1967), 1274-1276.

Cf. A055155 (m=1).

Cf. A000010, A000203, A001222, A008833, A010052, A034444, A061547.

R:= proc(n) uses numtheory; local K,k;
  K:= select(k -> (n mod k^2 = 0), divisors(n));
  add(k^2*2^nops(factorset(n/k^2)),k=K);
end proc:
seq(R(n),n=1..100); # Robert Israel, Oct 18 2015

a[n_]:=Total[GCD[#, n/#]^2 & /@ Divisors[n]]; Table[a[n], {n, 1, 87}] (* Jean-François Alcover, Jul 26 2011 *)
f[p_, e_] := If[OddQ[e], 2*(p^(e+1)-1)/(p^2-1), (p^(e+2)+p^e-2)/(p^2-1)]; a[1] = 1; a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100] (* Amiram Eldar, Sep 03 2020 *)

a(n) = Sum_{d divides n} gcd(d, n/d)^2. Multiplicative with a(p^e) = (p^(e+2)+p^e-2)/(p^2-1) if e is even else 2*(p^(e+1)-1)/(p^2-1). - Vladeta Jovovic, May 30 2003
Dirichlet g.f.: zeta^2(s)*zeta(2s-2)/zeta(2s). Dirichlet convolution of A034444 and the sequence n*A010052(n). - R. J. Mathar, Apr 18 2011
Inverse Mobius transform of A008833. - R. J. Mathar, Oct 31 2011
a(n) = Sum_{d divides n} (-1)^A001222(d) * A000010(d) * A000203(n/d) = Sum_{k^2 divides n} k^2 * 2^A001221(n/k^2). - Robert Israel, Oct 18 2015
Sum_{k=1..n} a(k) ~ Zeta(3/2)^2 * n^(3/2) / (3*Zeta(3)) - (3*n*(log(n) - 1 + 2*gamma + 2*log(2*Pi) - 12*Zeta'(2)/Pi^2))/Pi^2, where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, Feb 05 2019
a(2^k) = (3/2)*2^k + (1/6)*(-2)^k - 2/3 = A061547(k+2). - Amiram Eldar, Sep 03 2020

A244414 Remove highest power of 6 from n.

Original entry on oeis.org

1, 2, 3, 4, 5, 1, 7, 8, 9, 10, 11, 2, 13, 14, 15, 16, 17, 3, 19, 20, 21, 22, 23, 4, 25, 26, 27, 28, 29, 5, 31, 32, 33, 34, 35, 1, 37, 38, 39, 40, 41, 7, 43, 44, 45, 46, 47, 8, 49, 50, 51, 52, 53, 9, 55, 56, 57, 58, 59, 10, 61, 62, 63, 64, 65, 11

Offset: 1

Wolfdieter Lang, Jun 27 2014

This is instance g = 6 of the g-family of sequences, call it r(g,n), where for g >= 2 the highest power of g is removed from n. See the crossrefs.

The present sequence is not multiplicative: a(6) = 1 not a(2)*a(3) = 6. In the prime factor decomposition one has to consider a(2^e2*3^e^3) as one entity, also for e2 >= 0, e3 >= 0 with a(1) = 1, and apply the rule given in the formula section. With this rule the sequence will be multiplicative in an unusual sense. - Wolfdieter Lang, Feb 12 2018

			a(1) = 1 = 1/6^A122841(1) = 1/6^0.
a(9) = a(2^0*3^2), min(0,2) = 0, a(9) = 2^(0-0)*3^(2-0) = 1*9 = 9.
a(12) = a(2^2*3^1), m = min(2,1) = 1, a(12) = 2^(2-1)*3^(1-1) = 2^1*1 = 2.
a(30) = a(2*3*5) = a(2^1*3^1)*a(5) = 1*a(5) = 5.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A122841, A000265, A038502, A065883, A132739, A242603.

A007310, A007913, A008833 are used to express relationship between terms of this sequence.

a[n_] := n/6^IntegerExponent[n, 6]; Array[a, 66] (* Robert G. Wilson v, Feb 12 2018 *)

a(n) = n/6^valuation(n,6); \\ Joerg Arndt, Jun 28 2014

a(n) = n/6^A122841(n), n >= 1.
For n >= 2, a(n) is sort of multiplicative if a(2^e2*3^e3) = 2^(e2 - m)*3^(e3 - m) with m = m(e2, e3) = min(e2, e3), for e2, e3 >= 0, a(1) = 1, and a(p^e) = p^e for primes p >= 5.
From Peter Munn, Jun 04 2020: (Start)
Proximity to being multiplicative may be expressed as follows:
a(n * A007310(k)) = a(n) * a(A007310(k));
a(n^2) = a(n)^2;
a(n) = a(A007913(n)) * a(A008833(n)).
(End)
Sum_{k=1..n} a(k) ~ (3/7) * n^2. - Amiram Eldar, Nov 20 2022

Incorrect multiplicity claim corrected by Wolfdieter Lang, Feb 12 2018

A334748 Let p be the smallest odd prime not dividing the squarefree part of n. Multiply n by p and divide by the product of all smaller odd primes.

Original entry on oeis.org

3, 6, 5, 12, 15, 10, 21, 24, 27, 30, 33, 20, 39, 42, 7, 48, 51, 54, 57, 60, 35, 66, 69, 40, 75, 78, 45, 84, 87, 14, 93, 96, 55, 102, 105, 108, 111, 114, 65, 120, 123, 70, 129, 132, 135, 138, 141, 80, 147, 150, 85, 156, 159, 90, 165, 168, 95, 174, 177, 28, 183, 186, 189

Offset: 1

Peter Munn, May 09 2020

A permutation of A028983.
A007417 (which has asymptotic density 3/4) lists index n such that a(n) = 3n. The sequence maps the terms of A007417 1:1 onto A145204\{0}, defining a bijection between them.
Similarly, bijections are defined from the odd numbers (A005408) to the nonsquare odd numbers (A088828), from the positive even numbers (A299174) to A088829, from A003159 to the nonsquares in A003159, and from A325424 to the nonsquares in A036668. The latter two bijections are between sets where membership depends on whether a number's squarefree part divides by 2 and/or 3.

			84 = 21*4 has squarefree part 21 (and square part 4). The smallest odd prime absent from 21 = 3*7 is 5 and the product of all smaller odd primes is 3. So a(84) = 84*5/3 = 140.

Permutation of A028983.
Row 3, and therefore column 3, of A331590. Cf. A334747 (row 2).
A007913, A034386, A225546, A284723 are used in formulas defining the sequence.
The formula section details how the sequence maps the terms of A003961, A019565, A070826; and how f(a(n)) relates to f(n) for f = A008833, A048675, A267116; making use of A003986.
Subsequences: A016051, A145204\{0}, A329575.
Bijections are defined that relate to A003159, A005408, A007417, A036668, A088828, A088829, A299174, A325424.

a(n) = {my(c=core(n), m=n); forprime(p=3, , if(c % p, m*=p; break, m/=p)); m;} \\ Michel Marcus, May 22 2020

a(n) = n * p / (A034386(p-1)/2), where p = A284723(A007913(n)).
a(n) = A334747(A334747(n)).
a(n) = A331590(3, n) = A225546(4 * A225546(n)).
a(2*n) = 2 * a(n).
a(A019565(n)) = A019565(n+2).
a(k * m^2) = a(k) * m^2.
a(A003961(n)) = A003961(A334747(n)).
a(A070826(n)) = prime(n+1).
A048675(a(n)) = A048675(n) + 2.
A008833(a(n)) = A008833(n).
A267116(a(n)) = A267116(n) OR 1, where OR denotes the bitwise operation A003986.
a(A007417(n)) = A145204(n+1) = 3 * A007417(n).

cp's OEIS Frontend

A056192 a(n) = n divided by its characteristic cube divisor A056191.

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

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A334109 a(n) = A329697(A225546(n)).

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A334871 Number of steps needed to reach 1 when starting from n and iterating with A334870.

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A335324 Square part of 4th-power-free part of n.

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A366245 The largest infinitary divisor of n that is a term of A366243.

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A068976 a(n) = Sum_{d | n} d/core(d) where core(x) is the smallest number such that x*core(x) is a square.

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