A322008 -id:A322008 - cp's OEIS frontend

A306679 a(n) = round(1/(1-Integral_{x=0..1} f_n(x) dx)), where f_n is the n-th of all functions that are representable as x^x^...^x with m>=1 x's and parentheses inserted in all possible ways.

2, 5, 10, 2, 17, 6, 2, 4, 26, 13, 3, 5, 8, 2, 2, 4, 2, 37, 21, 8, 11, 15, 3, 5, 5, 6, 9, 6, 2, 2, 3, 2, 2, 4, 4, 2, 3, 50, 32, 16, 3, 19, 23, 7, 11, 2, 4, 7, 10, 12, 16, 13, 2, 3, 6, 3, 5, 5, 7, 6, 5, 9, 8, 6, 8, 2, 2, 2, 2, 2, 4, 2, 2, 2, 2, 2, 5, 4, 3, 4, 4

Offset: 1

Alois P. Heinz, Mar 04 2019

The ordering of the functions f_n is defined in A215703: f_1, f_2, ... = x, x^x, x^(x^2), x^(x^x), x^(x^3), x^(x^x*x), x^(x^(x^2)), x^(x^(x^x)), x^(x^4), x^(x^x*x^2), ... . Values of new records are in A322008.

Alois P. Heinz, Table of n, a(n) for n = 1..20299

Cf. A000081, A087803, A215703, A322008.

T:= proc(n) T(n):=`if`(n=1, [x], map(h-> x^h, g(n-1$2))) end:
g:= proc(n, i) option remember; `if`(i=1, [x^n], [seq(seq(
      seq(mul(T(i)[w[t]-t+1], t=1..j)*v, v=g(n-i*j, i-1)), w=
      combinat[choose]([$1..nops(T(i))+j-1], j)), j=0..n/i)])
    end:
a:= proc() local i, l; i, l:= 0, []; proc(n) while n>
      nops(l) do i:= i+1; l:= [l[], map(f-> round(evalf(
      1/(1-int(f, x=0..1)))), T(i))[]] od; l[n] end
    end():
seq(a(n), n=1..100);

a(n) >= 2 for n >= 1.

A322009 1/(Integral_{x=0..1} x^(x^(x^n)) dx - 1/2), rounded to the nearest integer.

Original entry on oeis.org

4, 14, 33, 64, 110, 174, 260, 369, 506, 672, 872, 1108, 1382, 1699, 2061, 2472, 2933, 3448, 4021, 4653, 5349, 6110, 6941, 7844, 8822, 9878, 11015, 12237, 13545, 14943, 16435, 18023, 19709, 21498, 23392, 25394, 27507, 29734, 32079, 34543, 37131, 39844, 42687, 45662, 48772

Offset: 0

M. F. Hasler, Mar 01 2019

nonn

Linked to the problem of sorting parenthesized expressions (x^x....^x) (cf. A000081 and A222379, A222380) according to the value of their integral from 0 to 1: This value is minimal, for a given number n of x's, for G[n](x) := x^((...(x^x)^x....)^x) = x^(x^(x^(n-2))), which converges pointwise to x^(x^0) = x^1 = x for all x in [0,1], as n -> oo. The corresponding integrals therefore tend to 1/2 as n -> oo. This sequence is a convenient measure of the distance of these integrals from 1/2.

See A322008 for the maximal values of such integrals.

			For n=0, Integral_{x=0..1} x^(x^(x^0)) dx = Integral_{x=0..1} x^x dx = A083648 = 0.7834..., and 1/(0.7834... - 0.5) = 1 / 0.2834... = 3.528..., so a(0) = round(3.528...) = 4.
For n=1, Integral_{x=0..1} x^(x^(x^1)) dx = Integral_{x=0..1} x^(x^x) dx = 0.5731..., and 1/(0.5731... - 0.5) = 1 / 0.0731... = 13.67..., so a(1) = round(13.67...) = 14.

Vladimir Reshetnikov, Integrals of power towers, on MathOverflow.net, Feb. 26, 2019.

Cf. A322008; A000081, A222379, A222380; A083648.

Digits:= 20:
a:= n-> round(evalf(1/(int(x^(x^(x^n)), x=0..1)-1/2))):
seq(a(n), n=0..44);  # Alois P. Heinz, Mar 01 2019

f[n_] := Round[1/(NIntegrate[x^(x^(x^n)), {x, 0, 1}, WorkingPrecision -> 24] - 1/2)]; Array[f, 45, 0] (* Robert G. Wilson v, Mar 01 2019 *)

PARI
```
A322009(n)=1\/intnum(x=0, 1, x^x^x^n-x)
```

cp's OEIS Frontend

A306679 a(n) = round(1/(1-Integral_{x=0..1} f_n(x) dx)), where f_n is the n-th of all functions that are representable as x^x^...^x with m>=1 x's and parentheses inserted in all possible ways.

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