A055076 -id:A055076 - cp's OEIS frontend

A358346 a(n) is the sum of the unitary divisors of n that are exponentially odd (A268335).

1, 3, 4, 1, 6, 12, 8, 9, 1, 18, 12, 4, 14, 24, 24, 1, 18, 3, 20, 6, 32, 36, 24, 36, 1, 42, 28, 8, 30, 72, 32, 33, 48, 54, 48, 1, 38, 60, 56, 54, 42, 96, 44, 12, 6, 72, 48, 4, 1, 3, 72, 14, 54, 84, 72, 72, 80, 90, 60, 24, 62, 96, 8, 1, 84, 144, 68, 18, 96, 144

Offset: 1

Amiram Eldar, Nov 11 2022

The number of unitary divisors of n that are exponentially odd is A055076(n).

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

Cf. A000290, A005117, A055076, A077610, A268335, A351569.

Similar sequences: A033634, A034448, A035316, A358347.

f[p_, e_] := 1 + If[OddQ[e], p^e, 0]; a[n_] := Times @@ f @@@ FactorInteger[n]; a[1] = 1; Array[a, 100]

a(n) = {my(f = factor(n)); prod(i = 1, #f~, 1 + if(f[i,2]%2,  f[i,1]^f[i,2], 0));}

a(n) >= 1 with equality if and only if n is a square (A000290).
a(n) <= A033634(n) with equality if and only if n is squarefree (A005117).
Multiplicative with a(p^e) = p^e + 1 if e is odd, and 1 otherwise.
a(n) = A034448(n)/A358347(n).
Sum_{k=1..n} a(k) ~ n^2/2.
From Amiram Eldar, Sep 14 2023: (Start)
a(n) = A034448(A350389(n)).
Dirichlet g.f.: zeta(s) * zeta(2*s-2) * Product_{p prime} (1 + 1/p^(s-1) - 1/p^(2*s-2) - 1/p^(2*s-1)). (End)

A368979 The number of exponential divisors of n that are exponentially odd numbers (A268335).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 1

Amiram Eldar, Jan 11 2024

First differs from A367516 at n = 128, and from A359411 at n = 512.

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

Cf. A001227, A008578, A049419, A138302, A268335, A322791, A368980.
Cf. A359411, A367516.
Similar sequences: A055076, A322483, A363825, A368977.

f[p_, e_] := DivisorSigma[0, e/2^IntegerExponent[e, 2]]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

a(n) = vecprod(apply(x -> numdiv(x >> valuation(x, 2)), factor(n)[, 2]));

Multiplicative with a(p^e) = A001227(e).
a(n) >= 1, with equality if and only if n is in A138302.
a(n) <= A049419(n), with equality if and only if n is noncomposite (A008578).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Product_{p prime} (1 + Sum_{k>=2} (d(k) - d(k-1))/p^k) = 1.13657098749361390865..., where d(k) is the number of odd divisors of k (A001227).

A367516 The number of unitary divisors of n that are exponentially evil numbers (A262675).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 1

Amiram Eldar, Nov 21 2023

First differs from A359411 at n = 128.

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

Cf. A010060, A262675, A270428, A359411, A367512.

Similar sequences: A034444, A055076, A056624, A366901, A366902, A367515.

f[p_, e_] := If[EvenQ[DigitCount[e, 2, 1]], 2, 1]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

a(n) = vecprod(apply(x -> 2-hammingweight(x)%2, factor(n)[, 2]));

from sympy import factorint
def A367516(n): return 1<Chai Wah Wu, Nov 23 2023

Multiplicative with a(p^e) = (2-A010060(e)).
a(n) = A034444(n)/A367515(n).
a(n) = 2^A367512(n).
a(n) >= 1, with equality if and only if n is an exponentially odious number (A270428).
a(n) <= A034444(n), with equality if and only if n is an exponentially evil number (A262675).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Product_{p prime} f(1/p) = 1.13071730542774788785..., where f(x) = 1/2 + x + ((1-x)/2) * Product_{k>=0} (1 - x^(2^k)).

A368168 The number of unitary divisors of n that are cubefull exponentially odd numbers (A335988).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 1

Amiram Eldar, Dec 14 2023

First differs from A359411 and A367516 at n = 64.

Also, the number of unitary divisors of the largest unitary divisor of n that is a cubefull exponentially odd number (A368167).

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

Cf. A013661, A034444, A055076, A325837, A335275, A335988, A368167, A368169.

Cf. A359411, A367516.

f[p_, e_] := If[e == 1 || EvenQ[e], 1, 2]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

a(n) = {my(f = factor(n)); prod(i = 1, #f~, if(f[i, 2] == 1 || !(f[i, 2]%2), 1, 2));}

a(n) = A034444(A368167(n)).
Multiplicative with a(p^e) = 2 if e is odd that is larger than 1, and 1 otherwise.
a(n) >= 1, with equality if and only if n is in A335275.
a(n) <= n, with equality if and only if n is in A335988.
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = zeta(2) * Product_{p prime} (1 - 1/p^2 + 1/p^3 - 1/p^4) = 1.12560687309375943599... .

A367515 The number of unitary divisors of n that are exponentially odious numbers (A270428).

Original entry on oeis.org

1, 2, 2, 2, 2, 4, 2, 1, 2, 4, 2, 4, 2, 4, 4, 2, 2, 4, 2, 4, 4, 4, 2, 2, 2, 4, 1, 4, 2, 8, 2, 1, 4, 4, 4, 4, 2, 4, 4, 2, 2, 8, 2, 4, 4, 4, 2, 4, 2, 4, 4, 4, 2, 2, 4, 2, 4, 4, 2, 8, 2, 4, 4, 1, 4, 8, 2, 4, 4, 8, 2, 2, 2, 4, 4, 4, 4, 8, 2, 4, 2, 4, 2, 8, 4, 4, 4

Offset: 1

Amiram Eldar, Nov 21 2023

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

Cf. A001285, A262675, A270428, A293439.

Similar sequences: A034444, A055076, A056624, A366901, A366902, A367516.

f[p_, e_] := If[OddQ[DigitCount[e, 2, 1]], 2, 1]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

a(n) = vecprod(apply(x -> hammingweight(x)%2+1, factor(n)[, 2]));

from sympy import factorint
def A367515(n): return 1<Chai Wah Wu, Nov 23 2023

Multiplicative with a(p^e) = A001285(e).
a(n) = A034444(n)/A367516(n).
a(n) = 2^A293439(n).
a(n) >= 1, with equality if and only if n is an exponentially evil number (A262675).
a(n) <= A034444(n), with equality if and only if n is an exponentially odious number (A270428).

A368883 The number of infinitary divisors of n that are cubefree.

Original entry on oeis.org

1, 2, 2, 2, 2, 4, 2, 3, 2, 4, 2, 4, 2, 4, 4, 1, 2, 4, 2, 4, 4, 4, 2, 6, 2, 4, 3, 4, 2, 8, 2, 2, 4, 4, 4, 4, 2, 4, 4, 6, 2, 8, 2, 4, 4, 4, 2, 2, 2, 4, 4, 4, 2, 6, 4, 6, 4, 4, 2, 8, 2, 4, 4, 2, 4, 8, 2, 4, 4, 8, 2, 6, 2, 4, 4, 4, 4, 8, 2, 2, 1, 4, 2, 8, 4, 4, 4

Offset: 1

Amiram Eldar, Jan 09 2024

The number of infinitary divisors of n that are squarefree is A055076(n).

Amiram Eldar, Table of n, a(n) for n = 1..10000
Vaclav Kotesovec, Graph - the asymptotic ratio (100000 terms)

Cf. A000583, A004709, A037445, A055076, A077609.

f[p_, e_] := Switch[Mod[e, 4], 1, 2, 2, 2, 3, 3, 0, 1]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

a(n) = vecprod(apply(x -> [1,2,2,3][x%4+1], factor(n)[, 2]));

Multiplicative with a(p^e) = 2 if e == 1 or 2 (mod 4), 3 if e == 3 (mod 4), and 1 if e == 0 (mod 4).
a(n) >= 1, with equality if and only if n is a 4th power (A000583).
a(n) <= A037445(n), with equality if and only if n is cubefree (A004709).
Dirichlet g.f.: zeta(4*s) * Product_{p prime} (1 + 2/p^s + 2/p^(2*s) + 3/p^(3*s)).
From Vaclav Kotesovec, Jan 09 2024: (Start)
Dirichlet g.f.: zeta(4*s) * zeta(s)^2 * Product_{p prime} (1 - 1/p^(2*s) + 1/p^(3*s) - 4/p^(4*s) + 3/p^(5*s)).
Let f(s) = Product_{p prime} (1 - 1/p^(2*s) + 1/p^(3*s) - 4/p^(4*s) + 3/p^(5*s)).
Sum_{k=1..n} a(k) ~ f(1) * zeta(4) * n * (log(n) + 2*gamma - 1 + f'(1)/f(1) + 4*zeta'(4)/zeta(4)), where
f(1) = Product_{p prime} (1 - 1/p^2 + 1/p^3 - 4/p^4 + 3/p^5) = 0.5857770602270641007515615375942370402509903724261557972367075945186871...,
f'(1) = f(1) * Sum_{p prime} (2*p^2 - p + 15) * log(p) / (p^4 + p^3 + p - 3) = f(1) * 1.319786264712492218167871116508220489817987315752197198819256094...,
gamma is the Euler-Mascheroni constant A001620, zeta(4) = Pi^4/90 = A013662 and for zeta'(4) see A261506. (End)

A368977 The number of bi-unitary divisors of n that are exponentially odd numbers (A268335).

Original entry on oeis.org

1, 2, 2, 1, 2, 4, 2, 3, 1, 4, 2, 2, 2, 4, 4, 3, 2, 2, 2, 2, 4, 4, 2, 6, 1, 4, 3, 2, 2, 8, 2, 4, 4, 4, 4, 1, 2, 4, 4, 6, 2, 8, 2, 2, 2, 4, 2, 6, 1, 2, 4, 2, 2, 6, 4, 6, 4, 4, 2, 4, 2, 4, 2, 3, 4, 8, 2, 2, 4, 8, 2, 3, 2, 4, 2, 2, 4, 8, 2, 6, 3, 4, 2, 4, 4, 4, 4

Offset: 1

Amiram Eldar, Jan 11 2024

Amiram Eldar, Table of n, a(n) for n = 1..10000
Vaclav Kotesovec, Graph - the asymptotic ratio (100000 terms)

Cf. A000005, A005117, A062503, A222266, A268335, A286324, A368978.

Similar sequences: A055076, A322483, A363825, A368979.

f[p_, e_] := If[OddQ[e], (e+3)/2, 2*Floor[e/4]+1]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

a(n) = vecprod(apply(x -> if(x%2, (x+3)/2, 2*(x\4)+1), factor(n)[, 2]));

for(n=1, 100, print1(direuler(p=2, n, (1 + X - X^2 + 2*X^3 - X^4)/(1 - X - X^4 + X^5))[n], ", ")) \\ Vaclav Kotesovec, Jan 11 2024

Multiplicative with a(p^e) = (e+3)/2 if e is odd, and 2*floor(e/4)+1 if e is even.
a(n) >= 1, with equality if and only if n is in A062503.
a(n) <= A000005(n), with equality if and only if n is squarefree (A005117).
From Vaclav Kotesovec, Jan 11 2024: (Start)
Dirichlet g.f.: zeta(s)^2 * Product_{p prime} (1 - (1 - p^s + 2*p^(2*s)) / (p^s*(1 + p^s)*(1 + p^(2*s)))).
Let f(s) = Product_{p prime} (1 - (1 - p^s + 2*p^(2*s)) / (p^s*(1 + p^s)*(1 + p^(2*s)))).
Sum_{k=1..n} a(k) ~ f(1) * n * (log(n) + 2*gamma - 1 + f'(1)/f(1)), where
f(1) = Product_{p prime} (1 - (1 - p + 2*p^2) / (p*(1 + p)*(1 + p^2))) = 0.5715031234451924252215041182933420817059774181158824297150124265420835...,
f'(1) = f(1) * Sum_{p prime} (4*p^5 - p^4 + 2*p^3 + 2*p + 1) * log(p) / (p^7 + 2*p^6 + p^5 + 3*p^4 + p^3 + p - 1) = f(1) * 1.1422556395248477875508983912036578244050011522937179465478688905880430...
and gamma is the Euler-Mascheroni constant A001620. (End)

A385042 The number of unitary divisors of n whose exponents in their prime factorizations are all powers of 2 (A138302).

Original entry on oeis.org

1, 2, 2, 2, 2, 4, 2, 1, 2, 4, 2, 4, 2, 4, 4, 2, 2, 4, 2, 4, 4, 4, 2, 2, 2, 4, 1, 4, 2, 8, 2, 1, 4, 4, 4, 4, 2, 4, 4, 2, 2, 8, 2, 4, 4, 4, 2, 4, 2, 4, 4, 4, 2, 2, 4, 2, 4, 4, 2, 8, 2, 4, 4, 1, 4, 8, 2, 4, 4, 8, 2, 2, 2, 4, 4, 4, 4, 8, 2, 4, 2, 4, 2, 8, 4, 4, 4

Offset: 1

Amiram Eldar, Jun 16 2025

First differs from A367515 at n = 128.

The sum of these divisors is A385043(n), and the largest of them is A367168(n).

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

The unitary analog of A353898.
Cf. A005117, A034444, A138302, A209229, A367168, A367515, A385043.
The number of unitary divisors of n that are: A000034 (power of 2), A055076 (exponentially odd), A056624 (square), A056671 (squarefree), A068068 (odd), A323308 (powerful), A365498 (cubefree), A365499 (biquadratefree), A368248 (cubefull), A380395 (cube), A382488 (3-smooth), this sequence (exponentially 2^n), A385044 (5-rough).

f[p_, e_] := Boole[e == 2^IntegerExponent[e, 2]] + 1; a[ 1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

PARI
```
a(n) = vecprod(apply(x -> (x == 1<
				
```

Multiplicative with a(p^e) = A209229(e) + 1.
a(n) <= A034444(n), with equality if and only if n is in A138302.
a(n) <= A353898(n), with equality if and only if n is squarefree (A005117).

A385044 The number of unitary divisors of n that are 5-rough numbers (A007310).

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 2, 1, 1, 2, 2, 1, 2, 2, 2, 1, 2, 1, 2, 2, 2, 2, 2, 1, 2, 2, 1, 2, 2, 2, 2, 1, 2, 2, 4, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 1, 4, 2, 2, 2, 2, 2, 2, 2, 2, 1, 4, 2, 2, 2, 2, 4, 2, 1, 2, 2, 2, 2, 4, 2, 2, 2, 1, 2, 2, 2, 4, 2, 2

Offset: 1

Amiram Eldar, Jun 16 2025

The sum of these divisors is A385045(n), and the largest of them is A065330(n).

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

The unitary analog of A035218.
Cf. A007310, A034444, A065330, A385045.
Cf. A001620, A013661, A306016.
The number of unitary divisors of n that are: A000034 (power of 2), A055076 (exponentially odd), A056624 (square), A056671 (squarefree), A068068 (odd), A323308 (powerful), A365498 (cubefree), A365499 (biquadratefree), A368248 (cubefull), A380395 (cube), A382488 (3-smooth), A385042 (exponentially 2^n), this sequence (5-rough).

f[p_, e_] := If[p <= 3, 1, 2]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

a(n) = vecprod(apply(x -> if(x <= 3, 1, 2), factor(n)[, 1]));

Multiplicative with a(p^e) = 1 if p <= 3, and 2 if p >= 5.
a(n) = A034444(n)/A382488(n).
a(n) <= A034444(n), with equality if and only if n is 5-rough.
a(n) <= A035218(n).
Dirichlet g.f.: (zeta(s)^2/zeta(2*s)) * (1/((1+1/2^s)*(1+1/3^s))).
Sum_{k=1..n} a(k) ~ (n / (2 * zeta(2))) *(log(n) + 2*gamma - 1 + log(2)/3 + log(3)/4 - 2*zeta'(2)/zeta(2)), where gamma is Euler's constant (A001620).

A327668 a(n) = n * Sum_{d|n} (-1)^(bigomega(d) - omega(d)) / d.

Original entry on oeis.org

1, 3, 4, 5, 6, 12, 8, 11, 11, 18, 12, 20, 14, 24, 24, 21, 18, 33, 20, 30, 32, 36, 24, 44, 29, 42, 34, 40, 30, 72, 32, 43, 48, 54, 48, 55, 38, 60, 56, 66, 42, 96, 44, 60, 66, 72, 48, 84, 55, 87, 72, 70, 54, 102, 72, 88, 80, 90, 60, 120, 62, 96, 88, 85, 84, 144, 68, 90, 96, 144

Offset: 1

Ilya Gutkovskiy, Sep 21 2019

Alois P. Heinz, Table of n, a(n) for n = 1..20000

Cf. A001221, A001222, A008836, A046660, A055076, A076479, A162511, A206369, A306408.

with(numtheory):
a:= n-> n*add((-1)^(bigomega(d)-nops(factorset(d)))/d, d=divisors(n)):
seq(a(n), n=1..80);  # Alois P. Heinz, Sep 21 2019

a[n_] := n Sum[(-1)^(PrimeOmega[d] - PrimeNu[d])/d, {d, Divisors[n]}]; Table[a[n], {n, 1, 70}]
f[p_, e_] := p^e + (p^e-(-1)^e)/(p+1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Dec 01 2022 *)

a(n) = {my(f = factor(n)); prod(i = 1, #f~, f[i,1]^f[i,2] + (f[i,1]^f[i,2] - (-1)^f[i,2])/(f[i,1]+1));} \\ Amiram Eldar, Dec 01 2022

G.f.: Sum_{k>=1} mu(rad(k)) * lambda(k) * x^k / (1 - x^k)^2.
a(p) = p + 1, where p is prime.
From Amiram Eldar, Dec 01 2022: (Start)
Multiplicative with a(p^e) = p^e + (p^e-(-1)^e)/(p+1).
Sum_{k=1..n} a(k) ~ c * n^2, where c = (1/2) * Product_{p prime} ((p^2+2)/(p^2+1)) = 0.7207673679... . (End)

cp's OEIS Frontend

A358346 a(n) is the sum of the unitary divisors of n that are exponentially odd (A268335).

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A368979 The number of exponential divisors of n that are exponentially odd numbers (A268335).

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A367516 The number of unitary divisors of n that are exponentially evil numbers (A262675).

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A368168 The number of unitary divisors of n that are cubefull exponentially odd numbers (A335988).

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A367515 The number of unitary divisors of n that are exponentially odious numbers (A270428).

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A368883 The number of infinitary divisors of n that are cubefree.

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A368977 The number of bi-unitary divisors of n that are exponentially odd numbers (A268335).

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A385042 The number of unitary divisors of n whose exponents in their prime factorizations are all powers of 2 (A138302).