A099353 -id:A099353 - cp's OEIS frontend

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

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

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 partial sums of A049419.
Cf. A099353, A327837.
Different from A013936 (which does not contain 52).

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

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

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

A099352 From P-positions in a certain game.

Original entry on oeis.org

0, 1, 3, 4, 5, 6, 8, 9, 10, 11, 13, 14, 15, 16, 17, 19, 20, 21, 22, 23, 24, 26, 27, 28, 29, 30, 31, 32, 33, 34, 36, 37, 38, 39, 40, 41, 42, 43, 44, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 69, 70, 71, 72, 73, 74, 75

Offset: 0

N. J. A. Sloane, Nov 16 2004

Nathaniel Johnston, Table of n, a(n) for n = 0..10000
A. S. Fraenkel, New games related to old and new sequences, INTEGERS, Electronic J. of Combinatorial Number Theory, Vol. 4, Paper G6, 2004.

Cf. A099353.

a:=proc(n) option remember: local j, t: if(n=0)then return 0: else t:=a(n-1)+1: for j from 0 to n-1 do if(t=b(j))then return t+1: elif(tNathaniel Johnston, Apr 28 2011

a[n_] := a[n] = Module[{j, t}, If[n == 0,  0,  t = a[n - 1] + 1; For[j = 0, j <= n - 1, j++, Which[t == b[j], Return[t + 1], t < b[j], Break[]]]; t]];
b[n_] := b[n] = If[n == 0, 0, b[n - 1] + a[n] - Floor[(a[n - 1] + 1)/a[n]] + 2];
Table[a[n], {n, 0, 70}] (* Jean-François Alcover, Mar 10 2023, after Nathaniel Johnston *)

Let a(n) = this sequence, b(n) = A099353. Then a(n) = the smallest number not in {a(0), b(0), a(1), b(1), ..., a(n-1), b(n-1)}; b(n) = b(n-1) + a(n) - floor((a(n-1)+1)/a(n)) + 2. Apart from initial zero, this is the complement of A099353.

cp's OEIS Frontend

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

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