A327837 -id:A327837 - cp's OEIS frontend

A099593 Sum of the number of e-divisors of all numbers from 2 up to n.

0, 1, 2, 4, 5, 6, 7, 9, 11, 12, 13, 15, 16, 17, 18, 21, 22, 24, 25, 27, 28, 29, 30, 32, 34, 35, 37, 39, 40, 41, 42, 44, 45, 46, 47, 51, 52, 53, 54, 56, 57, 58, 59, 61, 63, 64, 65, 68, 70, 72, 73, 75, 76, 78, 79, 81, 82, 83, 84, 86, 87, 88, 90, 94, 95, 96, 97, 99, 100, 101

Offset: 1

Ralf Stephan, Oct 26 2004

nonn

Amiram Eldar, Table of n, a(n) for n = 1..10000
J. Wu, Problème de diviseurs exponentiels et entiers exponentiellement sans facteur carré, J. Theor. Nombr. Bordeaux 7 (1) (1995) 133-141.

Equals - 1 + partial sums of A049419. Cf. A145353, A327837.

f[p_, e_]  := DivisorSigma[0, e]; ediv[n_] := Times @@ (f @@@ FactorInteger[n]);
Accumulate[Array[ediv, 100]] - 1 (* Amiram Eldar, Jun 23 2019 *)

a(n) ~ A327837 * n. - Vaclav Kotesovec, Feb 27 2023

A361063 Multiplicative with a(p^e) = sigma_2(e), where sigma_2 = A001157.

Original entry on oeis.org

1, 1, 1, 5, 1, 1, 1, 10, 5, 1, 1, 5, 1, 1, 1, 21, 1, 5, 1, 5, 1, 1, 1, 10, 5, 1, 10, 5, 1, 1, 1, 26, 1, 1, 1, 25, 1, 1, 1, 10, 1, 1, 1, 5, 5, 1, 1, 21, 5, 5, 1, 5, 1, 10, 1, 10, 1, 1, 1, 5, 1, 1, 5, 50, 1, 1, 1, 5, 1, 1, 1, 50, 1, 1, 5, 5, 1, 1, 1, 21, 21, 1, 1, 5

Offset: 1

Vaclav Kotesovec, Mar 01 2023

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

Cf. A001157, A049419, A361012, A361064.

Cf. A327837, A361013.

g[p_, e_] := DivisorSigma[2, e]; a[1] = 1; a[n_] := Times @@ g @@@ FactorInteger[n]; Array[a, 100]

a(n) = vecprod(apply(x -> sigma(x, 2), factor(n)[, 2])); \\ Amiram Eldar, Jan 07 2025

from math import prod
from sympy import factorint, divisor_sigma
def A361063(n): return prod(divisor_sigma(e,2) for e in factorint(n).values()) # Chai Wah Wu, Mar 01 2023

Dirichlet g.f.: Product_{primes p} (1 + Sum_{e>=1} sigma_2(e) / p^(e*s)).

Sum_{k=1..n} a(k) ~ c * n, where c = Product_{p prime} (1 + Sum_{e>=2} (sigma_2(e) - sigma_2(e-1)) / p^e) = 11.343154585178523783556367128387762286267199879648613456124659589127638983...

A361064 Multiplicative with a(p^e) = sigma_3(e), where sigma_3 = A001158.

Original entry on oeis.org

1, 1, 1, 9, 1, 1, 1, 28, 9, 1, 1, 9, 1, 1, 1, 73, 1, 9, 1, 9, 1, 1, 1, 28, 9, 1, 28, 9, 1, 1, 1, 126, 1, 1, 1, 81, 1, 1, 1, 28, 1, 1, 1, 9, 9, 1, 1, 73, 9, 9, 1, 9, 1, 28, 1, 28, 1, 1, 1, 9, 1, 1, 9, 252, 1, 1, 1, 9, 1, 1, 1, 252, 1, 1, 9, 9, 1, 1, 1, 73, 73, 1, 1, 9

Offset: 1

Vaclav Kotesovec, Mar 01 2023

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

Cf. A001158, A049419, A361012, A361063.

Cf. A327837, A361013.

g[p_, e_] := DivisorSigma[3, e]; a[1] = 1; a[n_] := Times @@ g @@@ FactorInteger[n]; Array[a, 100]

a(n) = vecprod(apply(x -> sigma(x, 3), factor(n)[, 2])); \\ Amiram Eldar, Jan 07 2025

from math import prod
from sympy import factorint, divisor_sigma
def A361064(n): return prod(divisor_sigma(e,3) for e in factorint(n).values()) # Chai Wah Wu, Mar 01 2023

Dirichlet g.f.: Product_{primes p} (1 + Sum_{e>=1} sigma_3(e) / p^(e*s)).

Sum_{k=1..n} a(k) ~ c * n, where c = Product_{p prime} (1 + Sum_{e>=2} (sigma_3(e) - sigma_3(e-1)) / p^e) = 136.775196585091127831467103699999450735835551529525277016916082455332230986...

A368541 The number of exponential divisors of the nonsquarefree numbers.

Original entry on oeis.org

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

Offset: 1

Amiram Eldar, Dec 29 2023

nonn

The terms of A049419 that are larger than 1, since A049419(k) = 1 if and only if k is squarefree (A005117).

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

Cf. A005117, A013929, A049419.
Cf. A084936, A174961, A275699, A368038, A368039, A368040.
Cf. A013661, A059956, A229099, A327837.

f[p_, e_] := DivisorSigma[0, e]; s[n_] := Times @@ f @@@ FactorInteger[n]; Select[Array[s, 200], # > 1 &]

lista(kmax) = {my(p, f); for(k = 1, kmax, f = factor(k); p = prod(i=1, #f~, numdiv(f[i, 2])); if(p > 1, print1(p, ", ")));}

a(n) = A049419(A013929(n)).

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = (A327837 - A059956)/A229099 = 2.53623753427906735929... .

A379714 Partial alternating sums of the number of exponential divisors function (A049419).

Original entry on oeis.org

1, 0, 1, -1, 0, -1, 0, -2, 0, -1, 0, -2, -1, -2, -1, -4, -3, -5, -4, -6, -5, -6, -5, -7, -5, -6, -4, -6, -5, -6, -5, -7, -6, -7, -6, -10, -9, -10, -9, -11, -10, -11, -10, -12, -10, -11, -10, -13, -11, -13, -12, -14, -13, -15, -14, -16, -15, -16, -15, -17, -16

Offset: 1

Amiram Eldar, Dec 30 2024

Amiram Eldar, Table of n, a(n) for n = 1..10000
László Tóth, Alternating Sums Concerning Multiplicative Arithmetic Functions, Journal of Integer Sequences, Vol. 20 (2017), Article 17.2.1. See section 4.15, p. 36.

Cf. A049419, A065442, A145353, A327837.

Similar sequences: A068762, A068773, A307704, A357817, A370895, A370896.

f[p_, e_]  := DivisorSigma[0, e]; ediv[n_] := Times @@ f @@@ FactorInteger[n]; Accumulate[Table[(-1)^(n+1)*ediv[n], {n, 1, 100}]]

ediv(n) = vecprod(apply(numdiv, factor(n)[, 2]));
list(nmax) = {my(s = 0); for(k = 1, nmax, s += (-1)^(k+1) * ediv(k); print1(s, ", "))};

a(n) = Sum_{k=1..n} (-1)^(k+1) * A049419(k).

Limit_{n->oo} a(n)/n = A327837 * (2/(A065442 + 1) - 1) = -0.37293122584744001729... .

cp's OEIS Frontend

A099593 Sum of the number of e-divisors of all numbers from 2 up to n.

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A361063 Multiplicative with a(p^e) = sigma_2(e), where sigma_2 = A001157.

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A361064 Multiplicative with a(p^e) = sigma_3(e), where sigma_3 = A001158.

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A368541 The number of exponential divisors of the nonsquarefree numbers.

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A379714 Partial alternating sums of the number of exponential divisors function (A049419).

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