|
%I #44 May 08 2022 15:42:41
%S 1,2,3,5,6,7,8,10,11,12,13,15,16,17,18,20,21,22,23,25,26,27,28,30,31,
%T 32,34,36,37,38,39,41,42,43,44,46,47,48,49,51,52,53,54,56,57,58,59,61,
%U 62,63,64,66,67,69,70,72,73,74,75,77,78,79,80,82,83,84,85,87,88,89,90
%N a(n) = [n/1] + [n/(2^2)] + [n/(3^3)] + [n/(4^4)] + ... + [n/(k^k)] + ..., up to infinity, where [ ] is the floor function.
%H Seiichi Manyama, <a href="/A062071/b062071.txt">Table of n, a(n) for n = 1..10000</a> (terms 1..1000 from Harry J. Smith)
%H Vaclav Kotesovec, <a href="/A062071/a062071.jpg">Plot of a(n)/n for n = 1..100000</a>
%F a(n) = Sum_{i=1..n} floor(n/i^i). - _Wesley Ivan Hurt_, Sep 15 2017
%F G.f.: (1/(1 - x)) * Sum_{k>=1} x^(k^k)/(1 - x^(k^k)). - _Seiichi Manyama_, Aug 30 2021
%F Conjecture: a(n) ~ c * n, where c = A073009. - _Vaclav Kotesovec_, Aug 30 2021
%e a(7) = [7/1] + [7/4] + [7/27] + ... = 7 + 1 + 0 + 0 + ... = 8.
%e a(8) = [8/1] + [8/4] + [8/27] + [8/256] + ... = 8 + 2 + 0 + 0 + ... = 10.
%t Flatten[{1, Table[Sum[Floor[n/k^k], {k, 1, Floor[N[Log[n]/LambertW[Log[n]]]] + 1}], {n, 2, 100}]}] (* _Vaclav Kotesovec_, Aug 30 2021 *)
%o (PARI) \p 10 v=[]; for(n=1,120,v=concat(v,suminf(k=1,floor(n/k^k)))); v
%o (PARI) for (n=1, 1000, write("b062071.txt", n, " ", suminf(k=1, n\k^k)\1) ) \\ _Harry J. Smith_, Jul 31 2009
%o (PARI) a(n)=sum(k=1,exp(lambertw(log(n)))+1,n\k^k) \\ _Charles R Greathouse IV_, May 28 2015
%o (SageMath) [sum( floor(n/j^j) for j in (1..1+log(n)) ) for n in (1..100)] # _G. C. Greubel_, May 06 2022
%Y Cf. A006218, A011371, A060832, A347397.
%K nonn,easy
%O 1,2
%A _Amarnath Murthy_, Jun 13 2001
%E More terms from _Jason Earls_, Jun 21 2001
|
