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

Author

Amarnath Murthy, Jun 13 2001

Keywords

• nonn
• easy

Examples

			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.
		

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

Crossrefs

Cf. A006218, A011371, A060832, A347397.

Programs

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 *)

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

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

Extensions

More terms from Jason Earls, Jun 21 2001