A062071 a(n) = [n/1] + [n/(2^2)] + [n/(3^3)] + [n/(4^4)] + ... + [n/(k^k)] + ..., up to infinity, where [ ] is the floor function.

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, 32, 34, 36, 37, 38, 39, 41, 42, 43, 44, 46, 47, 48, 49, 51, 52, 53, 54, 56, 57, 58, 59, 61, 62, 63, 64, 66, 67, 69, 70, 72, 73, 74, 75, 77, 78, 79, 80, 82, 83, 84, 85, 87, 88, 89, 90

Offset

1,2

Links

Seiichi Manyama, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Harry J. Smith)

Vaclav Kotesovec, Plot of a(n)/n for n = 1..100000

Formula

a(n) = Sum_{i=1..n} floor(n/i^i). - Wesley Ivan Hurt, Sep 15 2017

G.f.: (1/(1 - x)) * Sum_{k>=1} x^(k^k)/(1 - x^(k^k)). - Seiichi Manyama, Aug 30 2021

Conjecture: a(n) ~ c * n, where c = A073009. - Vaclav Kotesovec, Aug 30 2021

Example

a(7) = [7/1] + [7/4] + [7/27] + ... = 7 + 1 + 0 + 0 + ... = 8.
a(8) = [8/1] + [8/4] + [8/27] + [8/256] + ... = 8 + 2 + 0 + 0 + ... = 10.

Mathematica

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 *)

Prog

(PARI) \p 10 v=[]; for(n=1, 120, v=concat(v, suminf(k=1, floor(n/k^k)))); v
(PARI) for (n=1, 1000, write("b062071.txt", n, " ", suminf(k=1, n\k^k)\1) ) \\ Harry J. Smith, Jul 31 2009
(PARI) a(n)=sum(k=1, exp(lambertw(log(n)))+1, n\k^k) \\ Charles R Greathouse IV, May 28 2015
(SageMath) [sum( floor(n/j^j) for j in (1..1+log(n)) ) for n in (1..100)] # G. C. Greubel, May 06 2022

Crossrefs

Cf. A006218, A011371, A060832, A347397.

Sequence in context: A039145 A242491 A038129 * A087069 A023737 A037459

Adjacent sequences: A062068 A062069 A062070 * A062072 A062073 A062074

Keyword

nonn,easy

Author

Amarnath Murthy, Jun 13 2001

Extensions

More terms from Jason Earls, Jun 21 2001

Status

approved