A086858 -id:A086858 - cp's OEIS frontend

A095703 Integer part of the (positive) solution to x^x = 10^n.

1, 2, 3, 4, 5, 6, 7, 7, 8, 9, 10, 10, 11, 12, 12, 13, 13, 14, 15, 15, 16, 17, 17, 18, 18, 19, 19, 20, 21, 21, 22, 22, 23, 23, 24, 25, 25, 26, 26, 27, 27, 28, 28, 29, 29, 30, 30, 31, 31, 32, 32, 33, 33, 34, 34, 35, 35, 36, 36, 37, 37, 38, 38, 39, 39, 40, 40, 41, 41

Offset: 0

Cino Hilliard, Jul 06 2004

Alois P. Heinz, Table of n, a(n) for n = 0..10000 (first 1001 terms from Andrew Howroyd)

Cf. A066022, A086858.

a:= n-> `if`(n=0, 1, (t-> floor(t/LambertW(t)))(n*log(10))):
seq(a(n), n=0..80);  # Alois P. Heinz, Nov 07 2019

a(n) = floor(solve(x=1, max(10,n+1), x^x-10^n))

a(n) = {my(k=1); while(k^k <= 10^n, k++); k-1} \\ Andrew Howroyd, Nov 07 2019

a(n) = n*log(10)/LambertW(n*log(10)) for n > 0, a(0) = 1. - Alois P. Heinz, Nov 07 2019

Terms a(23) and beyond from Andrew Howroyd, Nov 07 2019

A328325 Expansion of Product_{k>=0} 1/(1 - x^(k^k)).

Original entry on oeis.org

1, 2, 3, 4, 6, 8, 10, 12, 15, 18, 21, 24, 28, 32, 36, 40, 45, 50, 55, 60, 66, 72, 78, 84, 91, 98, 105, 113, 122, 131, 140, 150, 161, 172, 183, 195, 208, 221, 234, 248, 263, 278, 293, 309, 326, 343, 360, 378, 397, 416, 435, 455, 476, 497, 519, 542, 566, 590, 615, 641, 668, 695

Offset: 0

Seiichi Manyama, Oct 12 2019

nonn

Partial sums of A328301.

			G.f.: 1/(1-x) + x/(1-x)^2 + x^4/((1-x)^2*(1-x^4)) + x^27/((1-x)^2*(1-x^4)*(1-x^27)) + ... .

Alois P. Heinz, Table of n, a(n) for n = 0..10000

Cf. A000312, A064985, A086858, A328301.

b:= proc(n, i) option remember; `if`(n=0 or i=1, 1,
      b(n, i-1)+(p-> `if`(p>n, 0, b(n-p, i)))(i^i))
    end:
a:= proc(n) option remember; `if`(n<2, n+1, a(n-1)+
      b(n, floor((t-> t/LambertW(t))(log(n)))))
    end:
seq(a(n), n=0..100);  # Alois P. Heinz, Oct 12 2019

b[n_, i_] := b[n, i] = If[n == 0 || i == 1, 1, b[n, i-1] + With[{p = i^i}, If[p > n, 0, b[n-p, i]]]];
a[n_] := a[n] = If[n < 2, n+1, a[n-1] + b[n, Floor[PowerExpand[Log[n]/ ProductLog[Log[n]]]]]];
Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Sep 14 2022, after Alois P. Heinz *)

N=99; x='x+O('x^N); m=1; while(N>=m^m, m++); Vec(1/prod(k=0, m-1, 1-x^k^k))

a(n) = Sum_{k=0..n} A328301(k).

G.f.: 1/(1-x) + Sum_{n>0} x^(n^n) / Product_{k=0..n} (1 - x^(k^k)).

cp's OEIS Frontend

A095703 Integer part of the (positive) solution to x^x = 10^n.

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A328325 Expansion of Product_{k>=0} 1/(1 - x^(k^k)).

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