A145353 Sum of the number of e-divisors of all numbers from 1 up to n.

1, 2, 3, 5, 6, 7, 8, 10, 12, 13, 14, 16, 17, 18, 19, 22, 23, 25, 26, 28, 29, 30, 31, 33, 35, 36, 38, 40, 41, 42, 43, 45, 46, 47, 48, 52, 53, 54, 55, 57, 58, 59, 60, 62, 64, 65, 66, 69, 71, 73, 74, 76, 77, 79, 80, 82, 83, 84, 85, 87, 88, 89, 91, 95, 96, 97, 98, 100, 101, 102, 103, 107

Offset: 1

Author

Jaroslav Krizek and N. J. A. Sloane, Mar 03 2009

Keywords

• nonn

Links

• 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.

Crossrefs

Equals partial sums of A049419.

Cf. A099353, A327837.

Different from A013936 (which does not contain 52).

Programs

Mathematica

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

PARI

d(n) = {my(f = factor(n)); prod(i = 1, #f~, numdiv(f[i,2]));}
lista(nmax) = {my(s = 0); for(n = 1, nmax, s += d(n); print1(s, ", ")); } \\ Amiram Eldar, Dec 08 2022

Formula

a(n) ~ c * n, where c = A327837. - Amiram Eldar, Dec 08 2022