A320111 -id:A320111 - cp's OEIS frontend

A355691 Dirichlet inverse of A320111, number of divisors of n that are not of the form 4k+2.

1, -1, -2, -1, -2, 2, -2, 0, 1, 2, -2, 2, -2, 2, 4, 1, -2, -1, -2, 2, 4, 2, -2, 0, 1, 2, 0, 2, -2, -4, -2, 1, 4, 2, 4, -1, -2, 2, 4, 0, -2, -4, -2, 2, -2, 2, -2, -2, 1, -1, 4, 2, -2, 0, 4, 0, 4, 2, -2, -4, -2, 2, -2, 0, 4, -4, -2, 2, 4, -4, -2, 0, -2, 2, -2, 2, 4, -4, -2, -2, 0, 2, -2, -4, 4, 2, 4, 0, -2, 2, 4, 2, 4, 2, 4, -2, -2, -1, -2, -1, -2, -4, -2, 0, -8

Offset: 1

Antti Karttunen, Jul 15 2022

Multiplicative because A320111 is.

Antti Karttunen, Table of n, a(n) for n = 1..100000

Cf. A320111.
Cf. also A355690.
Cf. A010892.

f[2, e_] := Switch[Mod[e, 6], 0, 0, 1, -1, 2, -1, 3, 0, 4, 1, 5, 1]; f[p_, 1] = -2; f[p_, 2] = 1; f[p_, e_] := 0; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Dec 30 2022 *)

A320111(n) = sumdiv(n,d,(2!=(d%4)));
memoA355691 = Map();
A355691(n) = if(1==n,1,my(v); if(mapisdefined(memoA355691,n,&v), v, v = -sumdiv(n,d,if(dA320111(n/d)*A355691(d),0)); mapput(memoA355691,n,v); (v)));

a(1) = 1, and for n > 1, a(n) = -Sum_{d|n, dA320111(n/d) * a(d).
Multiplicative with a(2^e) = A010892(e+2) and for a prime p > 2, a(p) = -2, a(p^2) = 1 and a(p^e) = 0 when e > 2. - Sebastian Karlsson, Oct 21 2022
Dirichlet g.f.: 4^s/(zeta(s)^2*(1 - 2^s + 4^s)). - Amiram Eldar, Dec 30 2022

A183063 Number of even divisors of n.

Original entry on oeis.org

0, 1, 0, 2, 0, 2, 0, 3, 0, 2, 0, 4, 0, 2, 0, 4, 0, 3, 0, 4, 0, 2, 0, 6, 0, 2, 0, 4, 0, 4, 0, 5, 0, 2, 0, 6, 0, 2, 0, 6, 0, 4, 0, 4, 0, 2, 0, 8, 0, 3, 0, 4, 0, 4, 0, 6, 0, 2, 0, 8, 0, 2, 0, 6, 0, 4, 0, 4, 0, 4, 0, 9, 0, 2, 0, 4, 0, 4, 0, 8, 0, 2, 0, 8, 0, 2

Offset: 1

Jaroslav Krizek, Dec 22 2010

Number of divisors of n that are divisible by 2. More generally, it appears that the sequence formed by starting with an initial set of k-1 zeros followed by the members of A000005, with k-1 zeros between every one of them, can be defined as "the number of divisors of n that are divisible by k", (k >= 1). For example if k = 1 we have A000005 by definition, if k = 2 we have this sequence. Note that if k >= 3 the sequences are not included in the OEIS because the usual OEIS policy is not to include sequences like this where alternate terms are zero; this is an exception. - Omar E. Pol, Oct 18 2011
Number of zeros in n-th row of triangle A247795. - Reinhard Zumkeller, Sep 28 2014
a(n) is also the number of partitions of n into equal parts, minus the number of partitions of n into consecutive parts. - Omar E. Pol, May 04 2017
a(n) is also the number of partitions of n into an even number of equal parts. - Omar E. Pol, May 14 2017

			For n = 12, set of even divisors is {2, 4, 6, 12}, so a(12) = 4.
On the other hand, there are six partitions of 12 into equal parts: [12], [6, 6], [4, 4, 4], [3, 3, 3, 3], [2, 2, 2, 2, 2, 2] and [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1]. And there are two partitions of 12 into consecutive parts: [12] and [5, 4, 3], so a(12) = 6 - 2 = 4, equaling the number of even divisors of 12. - _Omar E. Pol_, May 04 2017

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
Mircea Merca, Combinatorial interpretations of a recent convolution for the number of divisors of a positive integer, Journal of Number Theory, Volume 160, March 2016, Pages 60-75, function tau_e(n).

Cf. A000005, A000265, A001227, A007814, A000593, A183064, A136655, A125911, A320111.
Column 2 of A195050. - Omar E. Pol, Oct 19 2011
Cf. A001620, A027750, A083910, A244009.
Cf. A247795.

a183063 = sum . map (1 -) . a247795_row
-- Reinhard Zumkeller, Sep 28 2014, Jan 15 2013, Jan 10 2012

[IsOdd(n) select 0 else #[d:d in Divisors(n)|IsEven(d)]:n in [1..100]]; // Marius A. Burtea, Dec 16 2019

A183063 := proc(n)
    if type(n,'even') then
        numtheory[tau](n/2) ;
    else
        0;
    end if;
end proc: # R. J. Mathar, Jun 18 2015

Table[Length[Select[Divisors[n], EvenQ]], {n, 90}] (* Alonso del Arte, Jan 10 2012 *)
a[n_] := (e = IntegerExponent[n, 2]) * DivisorSigma[0, n / 2^e]; Array[a, 100] (* Amiram Eldar, Jul 06 2022 *)

a(n)=if(n%2,0,numdiv(n/2)) \\ Charles R Greathouse IV, Jul 29 2011

from sympy import divisor_count
def A183063(n): return divisor_count(n>>(m:=(~n&n-1).bit_length()))*m # Chai Wah Wu, Jul 16 2022

def A183063(n): return len([1 for d in divisors(n) if is_even(d)])
[A183063(n) for n in (1..80)]  # Peter Luschny, Feb 01 2012

a(n) = A000005(n) - A001227(n).
a(2n-1) = 0; a(2n) = A000005(n).
G.f.: Sum_{d>=1} x^(2*d)/(1 - x^(2*d)) and generally for the number of divisors that are divisible by k: Sum_{d>=1} x^(k*d)/(1 - x^(k*d)). - Geoffrey Critzer, Apr 15 2014
Dirichlet g.f.: zeta(s)^2/2^s and generally for the number of divisors that are divisible by k: zeta(s)^2/k^s. - Geoffrey Critzer, Mar 28 2015
From Ridouane Oudra, Sep 02 2019: (Start)
a(n) = Sum_{i=1..n} (floor(n/(2*i)) - floor((n-1)/(2*i))).
a(n) = 2*A000005(n) - A000005(2n). (End)
Conjecture: a(n) = lim_{x->n} f(Pi*x), where f(x) = sin(x)*Sum_{k>0} (cot(x/(2*k))/(2*k) - 1/x). - Velin Yanev, Dec 16 2019
a(n) = A000005(A000265(n))*A007814(n) - Chai Wah Wu, Jul 16 2022
Sum_{k=1..n} a(k) ~ n*log(n)/2 + (gamma - 1/2 - log(2)/2)*n, where gamma is Euler's constant (A001620). - Amiram Eldar, Nov 27 2022
Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k)/A000005(k) = 1-log(2) (A244009). - Amiram Eldar, Mar 01 2023

Formula corrected by Charles R Greathouse IV, Jul 29 2011

A152822 Periodic sequence [1,1,0,1] of length 4.

Original entry on oeis.org

1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1

Offset: 0

Richard Choulet, Dec 13 2008

Antti Karttunen, Table of n, a(n) for n = 0..65537
Index entries for characteristic functions.
Index entries for linear recurrences with constant coefficients, signature (0,0,0,1).

Characteristic function of A042965.
Cf. A026052, A026064, A320111 (inverse Möbius transform).
Sequence A166486 shifted by two terms.

a:= n-> [1,1,0,1][1+irem(n,4)]:
seq(a(n), n=0..104);  # Alois P. Heinz, Sep 01 2021

a(n)=n%4!=2 \\ Jaume Oliver Lafont, Mar 24 2009

A152822(n) = { my(f=factor(n)); prod(k=1,#f~,if(2==f[k,1],f[k,2]>1,1)); }; \\ (After multiplicative formula) - Antti Karttunen, May 03 2022

def A152822(n): return (1,1,0,1)[n&3] # Chai Wah Wu, Jan 10 2023

a(n) = 3/4 - (1/4)*(-1)^n + (1/2)*cos(n*Pi/2);
a(n+4) = a(n) with a(0) = a(1) = a(3) = 1 and a(2) = 0;
O.g.f.: (1+z+z^3)/(1-z^4);
a(n) = ceiling(cos(Pi*n/4)^2). - Wesley Ivan Hurt, Jun 12 2013
From Antti Karttunen, May 03 2022: (Start)
Multiplicative with a(p^e) = 1 for odd primes, and a(2^e) = [e > 1]. (Here [ ] is the Iverson bracket, i.e., a(2^e) = 0 if e=1, and 1 if e>1).
a(n) = A166486(2+n).
(End)
Dirichlet g.f.: zeta(s)*(1 - 1/2^s + 1/4^s). - Amiram Eldar, Dec 27 2022

More terms from Philippe Deléham, Dec 21 2008

Keyword:mult added by Andrew Howroyd, Jul 27 2018

A320107 a(n) = A001227(A252463(n)).

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 2, 1, 1, 2, 2, 2, 2, 2, 2, 1, 2, 3, 2, 2, 2, 2, 2, 2, 3, 2, 1, 2, 2, 4, 2, 1, 2, 2, 4, 3, 2, 2, 2, 2, 2, 4, 2, 2, 2, 2, 2, 2, 3, 3, 2, 2, 2, 4, 4, 2, 2, 2, 2, 4, 2, 2, 2, 1, 4, 4, 2, 2, 2, 4, 2, 3, 2, 2, 3, 2, 4, 4, 2, 2, 1, 2, 2, 4, 4, 2, 2, 2, 2, 6, 4, 2, 2, 2, 4, 2, 2, 3, 2, 3, 2, 4, 2, 2, 4

Offset: 1

Antti Karttunen, Nov 22 2018

nonn

Records 1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 20, 24, 32, 36, 40, ... occur at n = 1, 5, 18, 30, 90, 210, 450, 630, 1890, 3150, 5670, 6930, 20790, 34650, 62370, ...

Antti Karttunen, Table of n, a(n) for n = 1..16384
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537

Cf. A001227, A005940, A051064, A055457, A252463, A320106 (Möbius transform).

Cf. also A320111 and A319699, A319700, A319703, A319989.

A001227(n) = numdiv(n>>valuation(n, 2));
A064989(n) = {my(f); f = factor(n); if((n>1 && f[1,1]==2), f[1,2] = 0); for (i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f)};
A252463(n) = if(!(n%2),n/2,A064989(n));
A320107(n) = A001227(A252463(n));

A051064(n) = if(n<1, 0, 1+valuation(n, 3));
A320107(n) = if(n<=2, 1, my(p=A252463(n)); if(!(p%2), A320107(p), A051064(p)*A320107(p)));

A051064(n) = if(n<1, 0, 1+valuation(n, 3));
A055457(n) = if(n<1, 0, 1+valuation(n, 5));
A064989(n) = {my(f); f = factor(n); if((n>1 && f[1,1]==2), f[1,2] = 0); for (i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f)};
A320107(n) = if(n<=2, 1, if(!(n%4), A320107(n/2), if(2==(n%4), A051064(n)*A320107(n/2), if(!(n%3), A320107(A064989(n)), A055457(n)*A320107(A064989(n))))));

a(n) = A001227(A252463(n)).
a(1) = a(2) = 1; for n > 2, a(n) = a(n/2) when n == 0 mod 4, a(n) = A051064(n) * a(n/2) when n == 2 mod 4, a(n) = a(A064989(n)), when n == 3 mod 6, otherwise a(n) = A055457(n) * a(A064989(n)).
For n > 2, let p = A252463(n). If p is even, then a(n) = a(p), if p is odd, then a(n) = A051064(p) * a(p).

A069735 Number of regular orientable coverings of the Klein bottle with 2n lists.

Original entry on oeis.org

1, 3, 2, 5, 2, 6, 2, 7, 3, 6, 2, 10, 2, 6, 4, 9, 2, 9, 2, 10, 4, 6, 2, 14, 3, 6, 4, 10, 2, 12, 2, 11, 4, 6, 4, 15, 2, 6, 4, 14, 2, 12, 2, 10, 6, 6, 2, 18, 3, 9, 4, 10, 2, 12, 4, 14, 4, 6, 2, 20, 2, 6, 6, 13, 4, 12, 2, 10, 4, 12, 2, 21, 2, 6, 6, 10, 4, 12, 2, 18, 5, 6, 2, 20, 4, 6, 4, 14, 2, 18

Offset: 1

Valery A. Liskovets, Apr 07 2002

Dirichlet convolution of A000012 by A040001. - R. J. Mathar, Mar 30 2011

a(n) is the number of full-dimensional lattices with volume n in Z^2 which are symmetric about a coordinate axis (equivalently, about both). - Álvar Ibeas, Mar 19 2021

			x + 3*x^2 + 2*x^3 + 5*x^4 + 2*x^5 + 6*x^6 + 2*x^7 + 7*x^8 + 3*x^9 + 6*x^10 + ...

Andrey Zabolotskiy, Table of n, a(n) for n = 1..10000
Valery A. Liskovets and Alexander Mednykh, Number of non-orientable coverings of the Klein bottle, 2002.
John S. Rutherford, Sublattice enumeration. IV. Equivalence classes of plane sublattices by parent Patterson symmetry and colour lattice group type, Acta Cryst. (2009). A65, 156-163. [See Table 1]. - From _N. J. A. Sloane_, Feb 23 2009

Equals row sums of triangle A143110. - Gary W. Adamson, Jul 25 2008

Cf. A000005, A000012, A001227, A001620, A040001, A059841, A069734, A099774, A183063, A304182, A320111.

read("transforms") : nmax := 100 :
L := [1,1,seq(0,i=1..nmax)] :
MOBIUSi(%) :
MOBIUSi(%) ; # R. J. Mathar, Sep 25 2017
with(NumberTheory): seq(tau(n) + `if`(n::odd, 0, tau(n/2)), n=1..100); # Peter Luschny, Mar 19 2021

d[n_] := DivisorSigma[0, n];
a[n_] := If[EvenQ[n], d[n] + d[n/2], d[n]];
Array[a, 100] (* Jean-François Alcover, Aug 27 2019 *)

{a(n) = if( n<1, 0, numdiv(n) + if( n%2, 0, numdiv( n / 2)))} /* Michael Somos, Mar 24 2012 */

Multiplicative with a(2^e)=2e+1 and a(p^e)=e+1 for e>0 and an odd prime p.
a(n) = d(n)+d(n/2) for even n and a(n) = d(n) otherwise where d(n) is the number of divisors of n (A000005).
G.f.: Sum_{k>0} x^k*(1+2*x^k)/(1-x^(2*k)). - Vladeta Jovovic, Dec 16 2002
Dirichlet g.f.: (1+2^(-s))*zeta^2(s) [ Rutherford]. - N. J. A. Sloane, Feb 23 2009
Moebius transform is period 2 sequence [ 1, 2, ...]. - Michael Somos, Mar 24 2012
a(2*n - 1) = A099774(n).
a(n) = Sum_{ m: m^2|n } A304182(n/m^2). - Andrey Zabolotskiy, May 07 2018
Sum_{k=1..n} a(k) ~ 3*n*log(n)/2 + (3*gamma - 3/2 - log(2)/2)*n, where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, Oct 04 2019
a(n) = 3*tau(n) - tau(2*n). - Ridouane Oudra, Mar 15 2021
a(n) = A320111(n) + (A059841(n)*A000005(n)), i.e. a(n) = A320111(n) if n is odd, and a(n) = A320111(n) + A000005(n) if n is even. - Antti Karttunen, Mar 17 2021
a(n) = A000005(n) + A183063(n) = 2*A000005(n) - A001227(n). - Amiram Eldar, Dec 22 2023

Corrected by T. D. Noe, Nov 13 2006

A319703 a(n) = A003415(A252463(n)).

Original entry on oeis.org

0, 0, 1, 1, 1, 1, 1, 4, 4, 1, 1, 5, 1, 1, 5, 12, 1, 6, 1, 7, 7, 1, 1, 16, 6, 1, 12, 9, 1, 8, 1, 32, 9, 1, 8, 21, 1, 1, 13, 24, 1, 10, 1, 13, 16, 1, 1, 44, 10, 10, 15, 15, 1, 27, 10, 32, 19, 1, 1, 31, 1, 1, 24, 80, 14, 14, 1, 19, 21, 12, 1, 60, 1, 1, 21, 21, 12, 16, 1, 68, 32, 1, 1, 41, 16, 1, 25, 48, 1, 39, 16, 25, 31, 1

Offset: 1

Antti Karttunen, Nov 22 2018

nonn

Antti Karttunen, Table of n, a(n) for n = 1..16384

Cf. A003415, A252463.

Cf. also A319699, A319700, A319989, A320107, A320111.

A003415(n) = {my(fac); if(n<1, 0, fac=factor(n); sum(i=1, matsize(fac)[1], n*fac[i, 2]/fac[i, 1]))}; \\ From A003415
A064989(n) = {my(f); f = factor(n); if((n>1 && f[1,1]==2), f[1,2] = 0); for (i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f)};
A252463(n) = if(!(n%2),n/2,A064989(n));
A319703(n) = A003415(A252463(n));

a(n) = A003415(A252463(n)).

A319699 a(n) = A001065(A252463(n)).

Original entry on oeis.org

0, 0, 1, 1, 1, 1, 1, 3, 3, 1, 1, 6, 1, 1, 6, 7, 1, 4, 1, 8, 8, 1, 1, 16, 4, 1, 7, 10, 1, 9, 1, 15, 10, 1, 9, 21, 1, 1, 14, 22, 1, 11, 1, 14, 16, 1, 1, 36, 6, 6, 16, 16, 1, 13, 11, 28, 20, 1, 1, 42, 1, 1, 22, 31, 15, 15, 1, 20, 22, 13, 1, 55, 1, 1, 21, 22, 13, 17, 1, 50, 15, 1, 1, 54, 17, 1, 26, 40, 1, 33, 17, 26, 32, 1, 21, 76, 1, 8, 28, 43, 1, 21, 1, 46, 42

Offset: 1

Antti Karttunen, Nov 22 2018

nonn

Also for n > 1, sum of A318889(x) for all x encountered when map x -> A252463(x) is iterated, starting from x = A252463(n), until 1 is reached.

Antti Karttunen, Table of n, a(n) for n = 1..16384

Cf. A001065, A005940, A252463, A318889.

Cf. also A319694, A319700, A319703, A319989, A320107, A320111.

A001065(n) = (sigma(n)-n);
A064989(n) = {my(f); f = factor(n); if((n>1 && f[1,1]==2), f[1,2] = 0); for (i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f)};
A252463(n) = if(!(n%2),n/2,A064989(n));
A319699(n) = A001065(A252463(n));

a(n) = A001065(A252463(n)).

a(n) = A001065(n) - A318889(n).

A319700 a(n) = A051953(A252463(n)).

Original entry on oeis.org

0, 0, 1, 1, 1, 1, 1, 2, 2, 1, 1, 4, 1, 1, 4, 4, 1, 3, 1, 6, 6, 1, 1, 8, 3, 1, 4, 8, 1, 7, 1, 8, 8, 1, 7, 12, 1, 1, 12, 12, 1, 9, 1, 12, 8, 1, 1, 16, 5, 5, 14, 14, 1, 9, 9, 16, 18, 1, 1, 22, 1, 1, 12, 16, 13, 13, 1, 18, 20, 11, 1, 24, 1, 1, 12, 20, 11, 15, 1, 24, 8, 1, 1, 30, 15, 1, 24, 24, 1, 21, 15, 24, 30, 1, 19, 32, 1, 7, 16, 30, 1, 19, 1, 28, 22

Offset: 1

Antti Karttunen, Nov 22 2018

nonn

Antti Karttunen, Table of n, a(n) for n = 1..16384

Cf. A051953, A252463.

Cf. also A319699, A319703, A319989, A320107, A320111.

A051953(n) = (n - eulerphi(n));
A064989(n) = {my(f); f = factor(n); if((n>1 && f[1,1]==2), f[1,2] = 0); for (i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f)};
A252463(n) = if(!(n%2),n/2,A064989(n));
A319700(n) = A051953(A252463(n));

a(n) = A051953(A252463(n)).

A319989 a(n) = A303757(A252463(n)).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 3, 1, 1, 3, 2, 1, 2, 1, 3, 3, 1, 1, 4, 2, 1, 2, 3, 1, 1, 1, 2, 3, 1, 1, 4, 1, 1, 2, 3, 1, 2, 1, 2, 4, 1, 1, 4, 1, 1, 3, 3, 1, 2, 2, 4, 3, 1, 1, 5, 1, 1, 3, 2, 2, 2, 1, 3, 3, 1, 1, 5, 1, 1, 4, 3, 1, 2, 1, 4, 2, 1, 1, 6, 2, 1, 2, 3, 1, 3, 2, 2, 2, 1, 1, 5, 1, 2, 4, 4, 1, 1, 1, 4, 5

Offset: 1

Antti Karttunen, Nov 22 2018

nonn

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A252463, A303757.

Cf. also A319699, A319700, A319703, A320107, A320111.

Block[{s = Table[Which[n == 1, 1, EvenQ@n, n/2, True, Times @@ Power[Which[# == 1, 1, # == 2, 1, True, NextPrime[#, -1]] & /@ First@ #, Last@ #] &@ Transpose@ FactorInteger@ n], {n, 120}], t}, t = EulerPhi@ Range@ Max@ s; Map[Function[n, Count[t[[2 ;; n]], ?(# == t[[n]] &)]], s] /. 0 -> 1] (* _Michael De Vlieger, Nov 23 2018 *)

up_to = 65537;
ordinal_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), pt); for(i=1, length(invec), if(mapisdefined(om,invec[i]), pt = mapget(om, invec[i]), pt = 0); outvec[i] = (1+pt); mapput(om,invec[i],(1+pt))); outvec; };
Aux303757(n) = if(1==n,0,eulerphi(n));
v303757 = ordinal_transform(vector(up_to,n,Aux303757(n)));
A303757(n) = v303757[n];
A064989(n) = {my(f); f = factor(n); if((n>1 && f[1,1]==2), f[1,2] = 0); for (i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f)};
A252463(n) = if(!(n%2),n/2,A064989(n));
A319989(n) = A303757(A252463(n));

a(n) = A303757(A252463(n)).

A322805 Lexicographically earliest such sequence a that for all i, j, a(i) = a(j) => f(i) = f(j), where f(1)=0, f(2)=1, f(n)=-1 for odd primes, and f(n) = A252463(n) for any other number.

Original entry on oeis.org

1, 2, 3, 4, 3, 5, 3, 6, 6, 7, 3, 8, 3, 9, 8, 10, 3, 11, 3, 12, 12, 13, 3, 14, 11, 15, 10, 16, 3, 17, 3, 18, 16, 19, 17, 20, 3, 21, 22, 23, 3, 24, 3, 22, 14, 25, 3, 26, 27, 27, 28, 28, 3, 29, 24, 30, 31, 32, 3, 33, 3, 34, 23, 35, 36, 36, 3, 31, 37, 38, 3, 39, 3, 40, 20, 37, 38, 41, 3, 42, 18, 43, 3, 44, 41, 45, 46, 47, 3, 48, 49, 46, 50, 51, 52, 53, 3, 54, 30, 55, 3, 52, 3, 56, 33

Offset: 1

Antti Karttunen, Dec 26 2018

nonn

For all i, j:
  a(i) = a(j) => A319694(i) = A319694(j),
  a(i) = a(j) => A319699(i) = A319699(j),
  a(i) = a(j) => A319700(i) = A319700(j),
  a(i) = a(j) => A319703(i) = A319703(j),
  a(i) = a(j) => A319989(i) = A319989(j),
  a(i) = a(j) => A320110(i) = A320110(j) => A320111(i) = A320111(j).

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A252463.
Cf. A319694, A319699, A319700, A319703, A319989, A320110, A320111.
Cf. also A322807, A322809.

up_to = 65537;
rgs_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(om,invec[i]), my(pp = mapget(om, invec[i])); outvec[i] = outvec[pp] , mapput(om,invec[i],i); outvec[i] = u; u++ )); outvec; };
A064989(n) = {my(f); f = factor(n); if((n>1 && f[1,1]==2), f[1,2] = 0); for (i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f)};
A252463(n) = if(!(n%2),n/2,A064989(n));
A322805aux(n) = if(n<=2,n-1,if(isprime(n),-1,A252463(n)));
v322805 = rgs_transform(vector(up_to,n,A322805aux(n)));
A322805(n) = v322805[n];

cp's OEIS Frontend

A355691 Dirichlet inverse of A320111, number of divisors of n that are not of the form 4k+2.

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A183063 Number of even divisors of n.

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A152822 Periodic sequence [1,1,0,1] of length 4.

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A320107 a(n) = A001227(A252463(n)).

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A069735 Number of regular orientable coverings of the Klein bottle with 2n lists.

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A319703 a(n) = A003415(A252463(n)).

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A319699 a(n) = A001065(A252463(n)).

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