A322483 -id:A322483 - cp's OEIS frontend

A374537 a(n) is the sum of the squares of the divisors of n that are exponentially odd numbers.

1, 5, 10, 5, 26, 50, 50, 69, 10, 130, 122, 50, 170, 250, 260, 69, 290, 50, 362, 130, 500, 610, 530, 690, 26, 850, 739, 250, 842, 1300, 962, 1093, 1220, 1450, 1300, 50, 1370, 1810, 1700, 1794, 1682, 2500, 1850, 610, 260, 2650, 2210, 690, 50, 130, 2900, 850, 2810

Offset: 1

Amiram Eldar, Jul 11 2024

The number of divisors of n that are exponentially odd is A322483(n) and their sum is A033634(n).

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

Cf. A001157, A005117, A033634, A322483, A351265, A358346, A374458, A374538.

Cf. A002117, A013661, A065464, A183699.

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

a(n) = {my(f = factor(n), p = f[, 1], e = f[, 2]); prod(i = 1, #p, 1 + p[i]^2 * (p[i]^(4*((e[i]-1)\2)+4) - 1) / (p[i]^4 - 1));}

a(n) = A001157(n) if and only if n is squarefree (A005117).
Multiplicative with a(p^e) = 1 + p^2 * (p^(4*floor((e-1)/2)+4) - 1) / (p^4 - 1).
Dirichlet g.f.: zeta(s) * zeta(2*s-4) * Product_{p prime} (1 + 1/p^(s-2) - 1/p^(2*s-4)).
Sum_{k=1..n} a(k) = c * n^3 / 3, where c = zeta(2) * zeta(3) * Product_{p prime} (1 - 2/p^2 + 1/p^3) = A183699 * A065464 = 0.84677961058798544766... .

A384557 The number of exponential unitary (or e-unitary) 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, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1

Offset: 1

Amiram Eldar, Jun 03 2025

First differs from A359411 at n = 2097152 = 2^21: a(2097152) = 4, while A359411(2097152) = 2.
First differs from A368979 at n = 512 = 2^9: a(512) = 2, while A368979(512) = 3.
First differs from A367516 at n = 128 = 2^7: a(128) = 2, while A367516(128) = 1.
First differs from A382291 at n = 128 = 2^7: a(128) = 2, while A382291(128) = 4.
First differs from A368168 at n = 64 = 2^6: a(64) = 2, while A368168(64) = 1.
The sum of these divisors is A384559(n), and the largest of them is A331737(n).
The number of exponential unitary (or e-unitary) divisors of n is A278908(n) and the number of divisors of n that are exponentially odd numbers is A322483(n).
All the terms are powers of 2. The first term that is greater than 2 is a(32768) = 4.

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

Cf. A068068, A138302, A268335, A278908, A322483, A331737, A359411, A367516, A368168, A368979, A382291, A384559.

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

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

Multiplicative with a(p^e) = A068068(e).
a(n) >= 1, with equality if and only if n is in A138302.
a(n) <= A278908(n), with equality if and only if n is an exponentially odd number.
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) = 1.13551542615965557947..., where d(k) is the number of odd unitary divisors of k (A068068).

A332730 a(n) = Sum_{d|n} tau(d/gcd(d, n/d)), where tau = A000005.

Original entry on oeis.org

1, 3, 3, 5, 3, 9, 3, 8, 5, 9, 3, 15, 3, 9, 9, 11, 3, 15, 3, 15, 9, 9, 3, 24, 5, 9, 8, 15, 3, 27, 3, 15, 9, 9, 9, 25, 3, 9, 9, 24, 3, 27, 3, 15, 15, 9, 3, 33, 5, 15, 9, 15, 3, 24, 9, 24, 9, 9, 3, 45, 3, 9, 15, 19, 9, 27, 3, 15, 9, 27, 3, 40, 3, 9, 15

Offset: 1

Ilya Gutkovskiy, Feb 21 2020

Inverse Moebius transform of A322483.

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

Cf. A000005, A007425, A024206, A048691, A124315, A124316, A295316, A322483, A332619, A332713.

Table[Sum[DivisorSigma[0, d/GCD[d, n/d]], {d, Divisors[n]}], {n, 1, 75}]
f[p_, e_] := Floor[(e+3)/2]; A322483[n_] := If[n==1, 1, Times @@ (f @@@ FactorInteger[n])]; Table[Sum[A322483[d], {d, Divisors[n]}], {n, 1, 75}]
f[p_, e_] := Floor[(e + 1)*(e + 5)/4]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Dec 05 2022 *)

a(n) = Sum_{d|n} A322483(d).
a(n) = Sum_{d|n} tau(n/d) * A295316(d).
Multiplicative with a(p^e) = floor((e+1)*(e+5)/4) = A024206(e+2). - Amiram Eldar, Dec 05 2022

cp's OEIS Frontend

A374537 a(n) is the sum of the squares of the divisors of n that are exponentially odd numbers.

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A384557 The number of exponential unitary (or e-unitary) divisors of n that are exponentially odd numbers (A268335).

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A332730 a(n) = Sum_{d|n} tau(d/gcd(d, n/d)), where tau = A000005.

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