A206787 -id:A206787 - cp's OEIS frontend

A192066 Sum of the odd unitary divisors of n.

1, 1, 4, 1, 6, 4, 8, 1, 10, 6, 12, 4, 14, 8, 24, 1, 18, 10, 20, 6, 32, 12, 24, 4, 26, 14, 28, 8, 30, 24, 32, 1, 48, 18, 48, 10, 38, 20, 56, 6, 42, 32, 44, 12, 60, 24, 48, 4, 50, 26, 72, 14, 54, 28, 72, 8, 80, 30, 60, 24, 62, 32, 80, 1, 84, 48, 68, 18, 96, 48, 72, 10, 74, 38, 104, 20, 96, 56, 80, 6

Offset: 1

R. J. Mathar, Jun 22 2011

The unitary analog of A000593.

			n=9 has the divisors 1, 3 and 9, of which 3 is not a unitary divisor because gcd(3,9/3) = gcd(3,3) != 1. This leaves 1 and 9 as unitary divisors which sum to a(9) = 1+9 = 10.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
R. J. Mathar, Survey of Dirichlet series of multiplicative arithmetic functions, arXiv:1106.4038 [math.NT], 2011-2012, section 4.2.
Eric Weisstein's World of Mathematics, Unitary Divisor.
Wikipedia, Unitary divisor.

Cf. A068068, A034448.

Cf. A077610, A206787.

a192066 = sum . filter odd . a077610_row
-- Reinhard Zumkeller, Feb 12 2012

unitaryOddSigma := proc(n,k) local a,d ; a := 0 ; for d in numtheory[divisors](n) do if type(d,'odd') then if igcd(d,n/d) = 1 then a := a+d^k ; end if; end if; end do: a ; end proc:
A := proc(n) unitaryOddSigma(n,1) ;end proc:

a[n_] := DivisorSum[n, Boole[OddQ[#] && GCD[#, n/#] == 1]*#&];
Array[a, 80] (* Jean-François Alcover, Nov 16 2017 *)
f[2, p_] := 1; f[p_, e_] := p^e + 1; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Sep 18 2020 *)

a(n) = sumdiv(n, d, if ((gcd(d, n/d)==1) && (d%2), d)); \\ Michel Marcus, Nov 17 2017

a(n) = Sum_{d|n, d odd, gcd(d,n/d)=1} d.
Dirichlet g.f.: zeta(s)*zeta(s-1)*(1-2^(1-s))/( zeta(2s-1)*(1-2^(1-2s)) ).
Sum_{k=1..n} a(k) ~ Pi^2 * n^2 / (21*zeta(3)). - Vaclav Kotesovec, Feb 02 2019
Multiplicative with a(2^e) = 1, and a(p^e) = p^e + 1 for p > 2. - Amiram Eldar, Sep 18 2020

A327278 a(n) = Sum_{d|n, d odd} d * mu(d) * mu(n/d).

Original entry on oeis.org

1, -1, -4, 0, -6, 4, -8, 0, 3, 6, -12, 0, -14, 8, 24, 0, -18, -3, -20, 0, 32, 12, -24, 0, 5, 14, 0, 0, -30, -24, -32, 0, 48, 18, 48, 0, -38, 20, 56, 0, -42, -32, -44, 0, -18, 24, -48, 0, 7, -5, 72, 0, -54, 0, 72, 0, 80, 30, -60, 0, -62, 32, -24, 0, 84, -48, -68, 0, 96

Offset: 1

Ilya Gutkovskiy, Sep 15 2019

Dirichlet inverse of A000593.

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

Cf. A000593, A008683, A046692, A055615, A067856, A206787, A327276.

[&+[d*MoebiusMu(d)*MoebiusMu(n div d): d in [a:a in Divisors(n)| IsOdd(a)]]:n in [1..70]]; // Marius A. Burtea, Sep 15 2019

Table[DivisorSum[n, # MoebiusMu[#] MoebiusMu[n/#] &, OddQ[#] &], {n, 1, 69}]
a[n_] := If[n == 1, n, -Sum[If[d < n, DivisorSum[n/d, Mod[#, 2] # &] a[d], 0], {d, Divisors[n]}]]; Table[a[n], {n, 1, 69}]
f[p_, e_] := If[p == 2, -Boole[e == 1], Which[e == 1, -p - 1, e == 2, p, e > 2, 0]]; a[1] = 1; a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100] (* Amiram Eldar, Oct 25 2020 *)

G.f. A(x) satisfies: A(x) = x - Sum_{k>=2} A000593(k) * A(x^k).
Dirichlet g.f.: 1 / (zeta(s) * zeta(s-1) * (1 - 2^(1-s))).
a(1) = 1; a(n) = -Sum_{d|n, dA000593(n/d) * a(d).
a(n) = Sum_{d|n} A067856(n/d) * A055615(d).
Multiplicative with a(2^e) = -1 if e = 1 and 0 otherwise, and a(p^e) = -(p+1) if e = 1, p if e = 2 and 0 otherwise, for an odd prime p. - Amiram Eldar, Oct 25 2020
Sum_{k=1..n} abs(a(k)) ~ 30*n^2/Pi^4. - Vaclav Kotesovec, May 30 2024

A319697 Sum of even squarefree divisors of n.

Original entry on oeis.org

0, 2, 0, 2, 0, 8, 0, 2, 0, 12, 0, 8, 0, 16, 0, 2, 0, 8, 0, 12, 0, 24, 0, 8, 0, 28, 0, 16, 0, 48, 0, 2, 0, 36, 0, 8, 0, 40, 0, 12, 0, 64, 0, 24, 0, 48, 0, 8, 0, 12, 0, 28, 0, 8, 0, 16, 0, 60, 0, 48, 0, 64, 0, 2, 0, 96, 0, 36, 0, 96, 0, 8, 0, 76, 0, 40, 0, 112, 0, 12, 0, 84, 0, 64, 0, 88, 0, 24, 0, 48, 0, 48, 0, 96

Offset: 1

Antti Karttunen, Oct 31 2018

nonn

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

Cf. A048250, A146076, A183063, A206787, A319698.

Table[Total[Select[Divisors[n],EvenQ[#]&&SquareFreeQ[#]&]],{n,100}] (* Harvey P. Dale, May 18 2019 *)
f[2, e_] := 2; f[p_, e_] := p + 1; a[n_] := If[OddQ[n], 0, Times @@ f @@@ FactorInteger[n]]; Array[a, 100] (* Amiram Eldar, Jun 30 2022 *)

A319697(n) = sumdiv(n, d, (!(d%2))*issquarefree(d)*d);

a(n) = Sum_{d|n} A059841(d)*A008966(d)*d.

a(n) = A048250(n) - A206787(n).

A086761 Numbers k such that k-th cyclotomic polynomial has exactly 5 nonzero terms.

Original entry on oeis.org

5, 10, 20, 25, 40, 50, 80, 100, 125, 160, 200, 250, 320, 400, 500, 625, 640, 800, 1000, 1250, 1280, 1600, 2000, 2500, 2560, 3125, 3200, 4000, 5000, 5120, 6250, 6400, 8000, 10000, 10240, 12500, 12800, 15625, 16000, 20000, 20480, 25000, 25600, 31250, 32000

Offset: 1

Yuval Dekel (dekelyuval(AT)hotmail.com), Aug 02 2003

nonn

A206787(a(n)) = 6. - Reinhard Zumkeller, Feb 12 2012
All terms have the form 2^a 5^b with a >= 0 and b > 0. - T. D. Noe, Feb 13 2012
If the above holds for all terms then this sequence is 5 * A003592. - David A. Corneth, Jul 04 2018

Cf. A003592, A051664, A065119.

Select[Range[1000], Count[CoefficientList[Cyclotomic[#, x], x], 0] == EulerPhi[#] - 4 &] (* T. D. Noe, Feb 13 2012 *)

is(n) = v = Vec(polcyclo(n)); sum(i=1,#v,v[i]!=0) == 5 \\ David A. Corneth, Jul 04 2018

More terms from T. D. Noe, Feb 13 2012

A366887 Sum of the odd squarefree divisors permuted by A163511.

Original entry on oeis.org

1, 1, 1, 4, 1, 4, 4, 6, 1, 4, 4, 6, 4, 24, 6, 8, 1, 4, 4, 6, 4, 24, 6, 8, 4, 24, 24, 48, 6, 32, 8, 12, 1, 4, 4, 6, 4, 24, 6, 8, 4, 24, 24, 48, 6, 32, 8, 12, 4, 24, 24, 48, 24, 192, 48, 96, 6, 32, 32, 72, 8, 48, 12, 14, 1, 4, 4, 6, 4, 24, 6, 8, 4, 24, 24, 48, 6, 32, 8, 12, 4, 24, 24, 48, 24, 192, 48, 96, 6, 32, 32, 72, 8

Offset: 0

Antti Karttunen, Nov 04 2023

nonn

Antti Karttunen, Table of n, a(n) for n = 0..16384

Cf. A163511, A206787, A366888 (rgs-transform).

A163511(n) = if(!n, 1, my(p=2, t=1); while(n>1, if(!(n%2), (t*=p), p=nextprime(1+p)); n >>= 1); (t*p));
A206787(n) = sumdiv(n, d, d*issquarefree(2*d));
A366887(n) = A206787(A163511(n));

a(n) = A206787(A163511(n)).

A366888 Lexicographically earliest infinite sequence such that a(i) = a(j) => A366887(i) = A366887(j) for all i, j >= 0.

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 2, 3, 1, 2, 2, 3, 2, 4, 3, 5, 1, 2, 2, 3, 2, 4, 3, 5, 2, 4, 4, 6, 3, 7, 5, 8, 1, 2, 2, 3, 2, 4, 3, 5, 2, 4, 4, 6, 3, 7, 5, 8, 2, 4, 4, 6, 4, 9, 6, 10, 3, 7, 7, 11, 5, 6, 8, 12, 1, 2, 2, 3, 2, 4, 3, 5, 2, 4, 4, 6, 3, 7, 5, 8, 2, 4, 4, 6, 4, 9, 6, 10, 3, 7, 7, 11, 5, 6, 8, 12, 2, 4, 4, 6, 4, 9, 6, 10, 4, 9

Offset: 0

Antti Karttunen, Nov 04 2023

Restricted growth sequence transform of A366887.

Antti Karttunen, Table of n, a(n) for n = 0..65537

Cf. A163511, A206787, A366887.

Cf. also A366881.

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; };
A163511(n) = if(!n, 1, my(p=2, t=1); while(n>1, if(!(n%2), (t*=p), p=nextprime(1+p)); n >>= 1); (t*p));
A206787(n) = sumdiv(n, d, d*issquarefree(2*d));
A366887(n) = A206787(A163511(n));
v366888 = rgs_transform(vector(1+up_to,n,A366887(n-1)));
A366888(n) = v366888[1+n];

A284233 Sum of odd prime power divisors of n (not including 1).

Original entry on oeis.org

0, 0, 3, 0, 5, 3, 7, 0, 12, 5, 11, 3, 13, 7, 8, 0, 17, 12, 19, 5, 10, 11, 23, 3, 30, 13, 39, 7, 29, 8, 31, 0, 14, 17, 12, 12, 37, 19, 16, 5, 41, 10, 43, 11, 17, 23, 47, 3, 56, 30, 20, 13, 53, 39, 16, 7, 22, 29, 59, 8, 61, 31, 19, 0, 18, 14, 67, 17, 26, 12, 71, 12, 73, 37, 33, 19, 18, 16, 79, 5

Offset: 1

Ilya Gutkovskiy, Mar 23 2017

			a(15) = 8 because 15 has 4 divisors {1, 3, 5, 15} among which 2 are odd prime powers {3, 5} therefore 3 + 5 = 8.

Eric Weisstein's World of Mathematics, Prime Power.
Index entries for sequences related to sums of divisors.

Cf. A000961, A005069, A023888, A023889, A038712, A061345, A065091 (fixed points), A087436 (number of odd prime power divisors of n), A206787, A246655, A284117.

nmax = 80; Rest[CoefficientList[Series[Sum[Boole[PrimePowerQ[k] && Mod[k, 2] == 1] k x^k/(1 - x^k), {k, 1, nmax}], {x, 0, nmax}], x]]
Table[Total[Select[Divisors[n], PrimePowerQ[#] && Mod[#, 2] == 1 &]], {n, 80}]
f[p_, e_] := (p^(e + 1) - 1)/(p - 1) - 1; f[2, e_] := 0; a[1] = 0; a[n_] := Plus @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Jul 24 2024 *)

G.f.: Sum_{k>=1} A061345(k)*x^A061345(k)/(1 - x^A061345(k)).
a(n) = Sum_{d|n, d = p^k, p prime, p > 2, k > 0} d.
a(p^k) = p*(p^k - 1)/(p - 1) for p is a prime > 2.
a(2^k*p) = p for p is a prime > 2.
a(2^k) = 0.
Additive with a(2^e) = 0, and a(p^e) = (p^(e+1)-1)/(p-1) - 1 for an odd prime p. - Amiram Eldar, Jul 24 2024

cp's OEIS Frontend

A192066 Sum of the odd unitary divisors of n.

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A327278 a(n) = Sum_{d|n, d odd} d * mu(d) * mu(n/d).

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A319697 Sum of even squarefree divisors of n.

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A086761 Numbers k such that k-th cyclotomic polynomial has exactly 5 nonzero terms.

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A366887 Sum of the odd squarefree divisors permuted by A163511.

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A366888 Lexicographically earliest infinite sequence such that a(i) = a(j) => A366887(i) = A366887(j) for all i, j >= 0.

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A284233 Sum of odd prime power divisors of n (not including 1).

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