A098572 -id:A098572 - cp's OEIS frontend

A098573 Positive integers not appearing in sequence A098572, which calculates the values of floor(sum(m^(1/m),n=1..m)).

4, 8, 12, 17, 24, 32, 41, 52, 66, 82, 101, 124, 150, 181, 217, 259, 307, 362, 426, 500, 583, 679, 788, 911, 1051, 1209, 1387, 1588, 1814, 2067, 2351, 2668, 3022, 3418, 3858, 4347, 4891, 5494, 6162, 6902, 7719, 8622, 9618, 10715, 11923, 13252, 14711, 16314

Offset: 1

Mark Hudson (mrmarkhudson(AT)hotmail.com), Sep 16 2004

			floor(1^(1/1)+2^(1/2)+3^(1/3))=3, floor(1^(1/1)+2^(1/2)+3^(1/3)+4^(1/4))=5 and so 4 is a member of this sequence.

Cf. A098572.

ans:=[]: a:=0: last:=0: n:=1: do: a:=a+evalf(n^(1/n),50): if floor(a)-last>1 then ans:=[op(ans),floor(a)-1]: fi: last:=floor(a): n:=n+1: od:

A133497 Numbers k such that A098572(k) - A098572(k-1) = 2.

Original entry on oeis.org

4, 7, 10, 14, 20, 27, 35, 45, 58, 73, 91, 113, 138, 168, 203, 244, 291, 345, 408, 481, 563, 658, 766, 888, 1027, 1184, 1361, 1561, 1786, 2038, 2321, 2637, 2990, 3385, 3824, 4312, 4855, 5457, 6124, 6863, 7679, 8581, 9576, 10672, 11879, 13207, 14666, 16267, 18024, 19949

Offset: 1

Philippe Lallouet (philip.lallouet(AT)orange.fr), Dec 01 2007

			With b(k) = A098572(k): b(1) = 1, b(2) = 2, b(3) = 3, b(4) = 5, hence b(4)-b(3) = 2 and a(1) = 4.

Cf. A098572.

for n from 1 do
    if A098572(n)-A098572(n-1)= 2 then
        printf("%d,\n",n) ;
    end if;
end do: # R. J. Mathar, Mar 13 2015

(A=0);for(p=1,1000,B=A;A=B+p^(1/p);if(floor(A)-floor(B)-1;print(p)))

More terms from Hugo Pfoertner, Jul 04 2021

A329117 Decimal expansion of Sum_{k>=1} (k^(1/k^2) - 1).

Original entry on oeis.org

9, 7, 1, 4, 9, 9, 0, 3, 4, 2, 8, 3, 3, 0, 8, 7, 5, 7, 2, 2, 2, 6, 2, 5, 0, 6, 2, 3, 1, 4, 7, 5, 4, 5, 8, 0, 0, 2, 2, 5, 5, 1, 0, 1, 4, 8, 9, 7, 0, 2, 3, 9, 8, 4, 2, 9, 0, 8, 9, 0, 4, 2, 5, 5, 9, 4, 0, 8, 4, 1, 1, 7, 0, 0, 9, 9, 5, 5, 4, 2, 4, 3, 7, 3, 0

Offset: 0

Daniel Hoyt, Nov 05 2019

			0.971499034283308757222625062314754580022...

Cf. A073002, A098572.

digits = 120; d = 1; j = 2; s = Pi^2 * (2*Log[Glaisher] - Log[2*Pi]/6 - EulerGamma/6); While[Abs[d] > 10^(-digits - 5), d = (-1)^j/j!*Derivative[j][Zeta][2*j]; s += d; j++]; RealDigits[s, 10, 120][[1]] (* Vaclav Kotesovec, Jun 18 2023 *)

sumpos(k=1, k^(1/k^2) - 1) \\ Michel Marcus, Nov 05 2019

Equals Sum_{k>=1} (-1)^k / k! * k-th derivative of zeta(2*k). - Vaclav Kotesovec, Jun 18 2023

A363704 Decimal expansion of lim_{x -> infinity} ((Sum_{k>=1} (k^(1/k^(1 + 1/x)) - 1)) - x^2).

Original entry on oeis.org

9, 8, 8, 5, 4, 9, 6, 0, 1, 1, 4, 2, 2, 6, 8, 7, 5, 0, 6, 4, 4, 7, 5, 4, 1, 0, 8, 3, 3, 9, 9, 7, 1, 2, 6, 4, 4, 2, 1, 9, 9, 8, 6, 8, 3, 8, 0, 1, 5, 2, 3, 8, 8, 1, 7, 3, 5, 4, 3, 0, 7, 0, 6, 7, 9, 5, 2, 2, 3, 5, 4, 8, 4, 9, 2, 9, 2, 2, 1, 6, 2, 6, 9, 5, 3, 2, 6

Offset: 0

Daniel Hoyt, Jun 16 2023

That this constant is less than one allows Sum_{k>=1} (k^(1/k^(1 + 1/x)) - 1) = floor(x^2), when x is the square root of any natural number greater than 1.

The limit converges slowly.

			0.98854960114226875064475410833997126442199868380...

Daniel Hoyt, Computing the limiting difference between the sum and integral of x^(1/x), 2022, pp. 1-4.
Wikipedia, Stieltjes Constants.

Cf. A098572, A350862, A329117, A351885, A001620, A363716.

digits = 120; d = 1; j = 2; s = StieltjesGamma[1]; While[Abs[d] > 10^(-digits - 5), d = (-1)^j / j! * Derivative[j][Zeta][j]; s += d; j++]; RealDigits[s, 10, 120][[1]] (* Vaclav Kotesovec, Jun 17 2023 *)

# Gives 14 correct digits
from mpmath import stieltjes,fac
def limgen(n):
    terms = []
    for y in range(3, n):
        for x in range(y, n):
            terms.append((((-1)**y)*stieltjes(x)*(x-(y-1))**(y-2))/(fac(x-(y-2))*fac(y-2)))
    n,o_sum = 2,0
    while True:
        n_term = 1/((n-1)**(n+1))
        n_sum = o_sum + n_term
        if o_sum == n_sum:
            break
        o_sum = n_sum
        n += 1
    return sum(terms) + 0.5 - stieltjes(0) + n_sum
print(str(limgen(60))[:-1])

Equals 1/2 - A001620 + Sum_{k>=2} (1/(k-1)^(k+1)) + Sum_{k>=3} Sum_{n>=k} (((-1)^k)*Stieltjes(n)*(n-k+1)^(k-2))/((n-k+2)!*(k-2)!).
From Vaclav Kotesovec, Jun 17 2023: (Start)
Equals lim_{n->oo} (Sum_{m=1..n} m^(1/m)) - n - log(n)^2/2.
Equals sg1 + Sum_{k>=2} (-1)^k / k! * k-th derivative of zeta(k), where sg1 is the first Stieltjes constant (see A082633). (End)

A363716 Decimal expansion of Sum_{k>=2} (1/k!) * k-th derivative of zeta(k).

Original entry on oeis.org

9, 3, 6, 1, 9, 1, 3, 1, 9, 4, 0, 4, 4, 8, 7, 0, 5, 1, 6, 4, 1, 1, 9, 2, 0, 3, 4, 8, 0, 3, 1, 3, 4, 4, 8, 8, 2, 4, 7, 6, 7, 0, 6, 2, 7, 4, 0, 7, 2, 8, 3, 2, 7, 8, 8, 4, 3, 6, 1, 1, 9, 4, 5, 9, 9, 5, 8, 4, 7, 1, 7, 8, 9, 1, 6, 3, 7, 0, 0, 5, 6, 4, 9, 3, 3, 6, 3, 3, 8, 3, 9, 6, 2, 3, 6, 1, 1, 4, 5, 7, 2, 4, 8, 0, 2, 4

Offset: 0

Vaclav Kotesovec, Jun 17 2023

			0.9361913194044870516411920348031344882476706274072832788436119459958471789...

Robert G. Wilson v, Table of n, a(n) for n = 0..1000 (a(0)-a(105) from _Vaclav Kotesovec_)

Cf. A098572, A363704.

digits = 120; d = 1; j = 2; s = 0; While[Abs[d] > 10^(-digits - 5), d = 1 / j! * Derivative[j][Zeta][j]; s += d; j++]; RealDigits[s, 10, 120][[1]]

Equals lim_{n->oo} (Sum_{m=1..n} 1/m^(1/m)) - n + log(n)^2/2 + sg1, where sg1 is the first Stieltjes constant (see A082633).

cp's OEIS Frontend

A098573 Positive integers not appearing in sequence A098572, which calculates the values of floor(sum(m^(1/m),n=1..m)).

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A133497 Numbers k such that A098572(k) - A098572(k-1) = 2.

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A329117 Decimal expansion of Sum_{k>=1} (k^(1/k^2) - 1).

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A363704 Decimal expansion of lim_{x -> infinity} ((Sum_{k>=1} (k^(1/k^(1 + 1/x)) - 1)) - x^2).

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A363716 Decimal expansion of Sum_{k>=2} (1/k!) * k-th derivative of zeta(k).

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