A359699 - OEIS

%I #26 Nov 10 2023 10:42:37

%S 2,8,8,5,1,8,2,1,2,2,4,9,9,9,9,3,1,7,5,5,1,9,7,9,6,2,2,4,3,7,3,8,5,1,

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

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

%N Decimal expansion of x such that Gamma(t) and t^x*e^-t are tangent at one point.

%H Math Stack Exchange, <a href="https://math.stackexchange.com/questions/4497319/for-what-z-do-gammax-and-xze-x-share-a-tangent-line-at-exactly-one">For what z do Gamma(x) and x^z e^-x share a tangent line at exactly one point?</a>

%e 2.885182122499993175519796224373851235141...

%t x[t_] := t*(1 + PolyGamma[0, t]);

%t x[t] /. FindRoot[Gamma[t] == Exp[-t]*t^x[t], {t, 2}, WorkingPrecision -> 106] // RealDigits // First (* _Jean-François Alcover_, Mar 08 2023 *)

%o (PARI) apply(x->((x+lngamma(x))/log(x)), solve(x=2, 3, x*log(x)*(1+psi(x)) - (x+lngamma(x)))) \\ _Michel Marcus_, Mar 08 2023

%K cons,nonn

%O 1,1

%A _Jodi Spitz_, Mar 07 2023