A099353 -id:A099353

Search: a099353 -id:a099353

Displaying 1-2 of 2 results found.

page 1

A099352

From P-positions in a certain game.

+10
5

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,3

LINKS

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.

FORMULA

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.

MAPLE

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(t<b(j))then break: fi: od: return t: fi: end:

b:=proc(n) option remember: if(n=0)then return 0: else return b(n-1) + a(n) - floor((a(n-1)+1)/a(n)) + 2: fi: end:

seq(a(n), n=0..70); # Nathaniel Johnston, Apr 28 2011

MATHEMATICA

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 *)

CROSSREFS

Cf. A099353.

KEYWORD

nonn,easy

AUTHOR

N. J. A. Sloane, Nov 16 2004

STATUS

approved

A145353

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

+10
5

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

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.

FORMULA

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

MATHEMATICA

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

PROG

(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

CROSSREFS

Equals partial sums of A049419.

Cf. A099353, A327837.

Different from A013936 (which does not contain 52).

KEYWORD

nonn

AUTHOR

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

STATUS

approved

page 1

Search completed in 0.008 seconds