\documentclass[a4paper]{article}
\pagestyle{empty}         % no page numbers

% Title and author go into PDF (by hyperref pdfusetitle),
% but not actually displayed.
\title{Base 4/3 Sum of Digits}
\author{Kevin Ryde}
\date{March 2024}

\usepackage[T1]{fontenc}  % T1 for accents, before babel
\usepackage{amsmath}
\usepackage{hyphenat}     % for \hyp hyphenation of words with -
\usepackage[pdfusetitle,
            pdfsubject={OEIS A244041},
            pdfkeywords={OEIS, A244041, base 4/3, sum of digits, bound},
            pdflang={en}, % RFC3066 ISO639 code
            pdfborderstyle={/W 0},  % no border on hyperlinks
           ]{hyperref}

\usepackage{tikz}
\usetikzlibrary{calc}     % for ($(...)$) coordinate calculations
\tikzset{>=latex,         % arrowhead type
         font=\small}     % same as text, not the display \normalfont

% % Uncomment this to see a box around each page text width and height.
% \usepackage{fancybox}
% \fancypage{\setlength{\fboxsep}{0pt}\fbox}{}

% GP-DEFINE  read("interv.gp");
% GP-DEFINE  read("interv-wip.gp");
% GP-DEFINE  nearly_equal_epsilon = 1e-15;
% GP-DEFINE  nearly_equal(x,y, epsilon=nearly_equal_epsilon) = \
% GP-DEFINE    abs(x-y) < epsilon;
% GP-DEFINE  const_log43 = log(4/3);
% GP-DEFINE  log43(x) = log(x) / const_log43;


%------------------------------------------------------------------------------
\begin{document}
\hfil {\large
       A244041 Base 4/3 \kern.1em Sum of Digits}
\hfil

\vspace{.5ex}

\hfil {Kevin Ryde, March 2024}
\hfil

\newcommand\MyTightDots{.\kern.08em.\kern.08em.}

\vspace{\baselineskip}

\href{http://oeis.org/A244041}{A244041} is the sum of digits of $n$ in
fractional base $4/3$.  Charles Greathouse in a comment gives the
following bounds, and wonders whether factor $11$ might be reduced to
$7$ or $8$.
\begin{align}
a(n)                        \mskip1mu
    <                       \mskip2mu
      3 \log_{4/3} n        \mskip2mu
    <                       \mskip1mu
      11 \log n             \kern2.5em
    \text{for $\thickmuskip=\thinmuskip % around "\ge" operator
               n \ge 2$}
% GP-Test  my(n=1); nearly_equal(log43(n), 0)
% GP-Test  my(n=2); nearly_equal(log43(n), 2.409, 1e-3)
\notag
\end{align}

$3 \log_{4/3} n$ would be every digit $3$, with the $\log$ as a proxy
for the number of base $4/3$ digits in $n$.  The $\log$ exceeds that
number since any
 $\thickmuskip=\medmuskip % around "\ge" operator
  n \ge 3$
starts with a digit $3$ so the smallest number with $k$ digits is at
least
 $n = 3 . (\tfrac43)^{k-1}$
which has
 $\log_{4/3} n = k + 2.818 \MyTightDots$
%
% GP-Test  my(k=10, n=3*(4/3)^(k-1)); \
% GP-Test  nearly_equal(log43(n), k + 2.818, 1e-3)
% GP-Test  nearly_equal(-1 + log43(3), 2.818, 1e-3)
% GP-Test  my(k=20); 3*(4/3)^(k-1) == 9/4*(4/3)^k
%
 \kern.2em
Factor $11$ is \kern.1em
 $3 / \log(4/3) = 10.428\MyTightDots$
rounded up.
% GP-Test  nearly_equal(3 / log(4/3), 10.428, 1e-3)
% 3 / log(4/3)
% not in OEIS: 10.428178490
% cf A083679 = log(4/3)

One way to experiment with what factor might be needed is to take a
sum of digits $s$ and find the smallest $n$ where sum
 $\thickmuskip=\medmuskip % around "=" operator
  a(n) = s$
 \hskip.03em plus .03em minus .02em
occurs.  This is
 $\thickmuskip=\medmuskip % around "=" operator
  n = \mathrm{A357425}(s)$
and the following plot is the factor needed to cover $s$.

% GP-DEFINE  read("OEIS-data.gp");
% GP-DEFINE  A357425 = OEIS_data_func("A357425",'bfile);
% GP-DEFINE  max_s = #OEIS_data("A357425",'bfile) - 1;
% GP-DEFINE  bound_of_s(s) = s / log(A357425(s)); \\ not exact

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    --(350,7.641622)--(351,7.641622)--(351,7.615621)--(352,7.615621)
    --(352,7.637318)--(353,7.637318)--(353,7.611505)--(354,7.611505)
    --(354,7.633068)--(355,7.633068)--(355,7.60744)--(356,7.60744)
    --(356,7.62887)--(357,7.62887)--(357,7.603425)--(358,7.603425)
    --(358,7.624723)--(359,7.624723)--(359,7.646021)--(360,7.646021)
    --(360,7.620627)--(361,7.620627)--(361,7.641796)--(362,7.641796)
    --(362,7.616581)--(363,7.616581)--(363,7.637621)--(364,7.637621)
    --(364,7.612583)--(365,7.612583)--(365,7.633497)--(366,7.633497)
    --(366,7.65441)--(367,7.65441)--(367,7.629422)--(368,7.629422)
    --(368,7.65021)--(369,7.65021)--(369,7.670999)--(370,7.670999)
    --(370,7.64606)--(371,7.64606)--(371,7.666725)--(372,7.666725)
    --(372,7.68739)--(373,7.68739)--(373,7.662502)--(374,7.662502)
    --(374,7.637906)--(375,7.637906)--(375,7.613598)--(376,7.613598)
    --(376,7.633901)--(377,7.633901)--(377,7.609757)--(378,7.609757)
    --(378,7.629942)--(379,7.629942)--(379,7.650127)--(380,7.650127)
    --(380,7.626028)--(381,7.626028)--(381,7.602207)--(382,7.602207)
    --(382,7.62216)--(383,7.62216)--(383,7.642113)--(384,7.642113)
    --(384,7.618336)--(385,7.618336)--(385,7.603782)--(386,7.603782)
    --(386,7.623532)--(387,7.623532)--(387,7.634282)--(388,7.634282)
    --(388,7.619739)--(389,7.619739)--(389,7.639378)--(390,7.639378)
    --(390,7.615988)--(391,7.615988)--(391,7.626626)--(392,7.626626)
    --(392,7.646131)--(393,7.646131)--(393,7.631699)--(394,7.631699)
    --(394,7.642259)--(395,7.642259)--(395,7.61914)--(396,7.61914)
    --(396,7.638429)--(397,7.638429)--(397,7.657718)--(398,7.657718)
    --(398,7.634642)--(399,7.634642)--(399,7.653824)--(400,7.653824)
    --(400,7.630896)--(401,7.630896)--(401,7.616845)--(402,7.616845)
    --(402,7.627191)--(403,7.627191)--(403,7.613232)--(404,7.613232)
    --(404,7.623526)--(405,7.623526)--(405,7.601133)--(406,7.601133)
    --(406,7.619901)--(407,7.619901)--(407,7.638669)--(408,7.638669)
    --(408,7.616315)--(409,7.616315)--(409,7.634983)--(410,7.634983)
    --(410,7.612767)--(411,7.612767)--(411,7.631335)--(412,7.631335)
    --(412,7.613974)--(413,7.613974)--(413,7.627726)--(414,7.627726)
    --(414,7.646195)--(415,7.646195)--(415,7.624155)--(416,7.624155)
    --(416,7.642527)--(417,7.642527)--(417,7.660898)--(418,7.660898)
    --(418,7.638897)--(419,7.638897)--(419,7.657172)--(420,7.657172)
    --(420,7.635305)--(421,7.635305)--(421,7.653485)--(422,7.653485)
    --(422,7.631751)--(423,7.631751)--(423,7.649836)--(424,7.649836)
    --(424,7.66792)--(425,7.66792)--(425,7.646225)--(426,7.646225)
    --(426,7.664216)--(427,7.664216)--(427,7.682207)--(428,7.682207)
    --(428,7.660549)--(429,7.660549)--(429,7.678448)--(430,7.678448)
    --(430,7.65692)--(431,7.65692)--(431,7.635612)--(432,7.635612)
    --(432,7.653328)--(433,7.653328)--(433,7.632146)--(434,7.632146)
    --(434,7.649772)--(435,7.649772)--(435,7.628715)--(436,7.628715)
    --(436,7.646252)--(437,7.646252)--(437,7.625319)--(438,7.625319)
    --(438,7.642768)--(439,7.642768)--(439,7.660217)--(440,7.660217)
    --(440,7.639318)--(441,7.639318)--(441,7.656681)--(442,7.656681)
    --(442,7.674043)--(443,7.674043)--(443,7.653179)--(444,7.653179)
    --(444,7.670455)--(445,7.670455)--(445,7.68773)--(446,7.68773)
    --(446,7.666902)--(447,7.666902)--(447,7.684093)--(448,7.684093)
    --(448,7.663385)--(449,7.663385)--(449,7.680491)--(450,7.680491)
    --(450,7.659902)--(451,7.659902)--(451,7.676924)--(452,7.676924)
    --(452,7.693946)--(453,7.693946)--(453,7.673392)--(454,7.673392)
    --(454,7.662167)--(455,7.662167)--(455,7.669894)--(456,7.669894)
    --(456,7.686751)--(457,7.686751)--(457,7.66643)--(458,7.66643)
    --(458,7.683206)--(459,7.683206)--(459,7.699981)--(460,7.699981)
    --(460,7.679694)--(461,7.679694)--(461,7.696389)--(462,7.696389)
    --(462,7.713084)--(463,7.713084)--(463,7.692832)--(464,7.692832)
    --(464,7.709447)--(465,7.709447)--(465,7.689308)--(466,7.689308)
    --(466,7.705844)--(467,7.705844)--(467,7.72238)--(468,7.72238)
    --(468,7.702276)--(469,7.702276)--(469,7.691266)--(470,7.691266)
    --(470,7.671473)--(471,7.671473)--(471,7.687795)--(472,7.687795)
    --(472,7.695239)--(473,7.695239);
  \coordinate (x-min) at (0,0 |- y-min);
  \coordinate (x-max) at (473,0 |- y-min);
  % end generated

  % X axis
  \draw[dashed,->] (x-min) -- ($(x-max) + (1, 0)$)
    node[at end,above left,yshift=.5ex] {$s$};
  \coordinate (x-tick) at (0,-.012);
  \foreach \x in {3, 50, 100, ...,450} {
    \draw ($(\x,0) + (y-min)$) -- +(x-tick)
      +(.5,0) node[at end,below] {\x};
  }

  % Y axis
  \draw[->] (y-min) -- ($(y-max) + (0,.18)$)
    node[at end,below right,xshift=1em,yshift=-2ex]
        {$\dfrac{s}{\log \mathrm{A357425}(s)}$
         \kern.4em bound factor};
  \coordinate (y-tick) at (-2,0);
  \foreach \y in {2.0, 3.0, 4.0, 5.0, 6.0, 7.0, 8.0} {
    \draw (0,\y) -- +(y-tick)
      node[at end,left=.1em,inner xsep=0pt] {\y};
  }

  \coordinate (seven) at (156.5, 6.9);
  \draw[densely dashed] ($(seven) +(0,-.33)$) -- (seven)
    node[at start,below,align=left,xshift=.3em]
        {$s {=} 156$ \\
         \kern.05em
         $7.009 \MyTightDots$};
% 7.009992
% GP-Test  my(s=156, r=bound_of_s(s)); print(r); \
% GP-Test  nearly_equal(r, 7.009, 1e-3)

  % % n=372 previous peak
  % \coordinate (earlier-peak) at (372.5, 7.55);
  % \draw[densely dashed] ($(earlier-peak) +(-10,-.33)$)
  %   .. controls +(40:.1) and +(-90:.22) .. (earlier-peak)
  %   node[at start,below,align=left,xshift=-.1em]
  %       {$\kern.1em s {=} 372$ \\
  %        $7.6873 \MyTightDots$};
% GP-Test  my(s=372, r=bound_of_s(s)); \
% GP-Test  nearly_equal(r, 7.6873, 1e-4)

  % n=467 peak
  \coordinate (peak) at (467.5, 7.65);
  \draw[densely dashed] ($(peak) +(-18,-.33)$)
    .. controls +(45:.1) and +(-90:.22) .. (peak)
    node[at start,below,align=left,xshift=-.3em,
         inner ysep=.2ex]
        {peak at \\
         $s {=} 467$ \\
         $7.722 \MyTightDots$};
% GP-DEFINE  peak_s = 467;
% GP-Test  my(r=bound_of_s(peak_s)); \
% GP-Test   print(r); \
% GP-Test   nearly_equal(r, 7.722, 1e-3)
% GP-DEFINE  bound_of_s_interv(s) = {
% GP-DEFINE    my(f=A357425(s));
% GP-DEFINE    interv_div(s, interv_log([f,f],3000));
% GP-DEFINE  }
% bound_of_s_interv(peak_s)*1.0
% interv_log([1000,1000],5000)*1.0
% interv_log_1plusx([1/10,1/10],10)*1.0
% s=467
% d=interv_div(A357425(s),interv_exp([58,58],100))*1.0
% interv_log(d,100)*1.0
% log(A357425(s))

\end{tikzpicture}
\end{center}

Prospective factor $7$ is surpassed at
 $\thickmuskip=\thinmuskip % around "=" operator
  s = 156$
where the plot is still rising, though at a slowing rate.
% GP-Test  nearly_equal(156 / log(4622358439), 7.00992, 1e-5)

The peak so far is near the end
 $\thickmuskip=.5\thinmuskip % around "=" operator
                 s = 467$
% GP-Test  peak_s == 467
% GP-Test  my(v=vector(max_s,s, if(s>=2,bound_of_s(s)))); \
% GP-Test  my(pos); vecmax(v,&pos); pos == peak_s
with
 $\thickmuskip=.5\thinmuskip % around "=" operator
                        n = 183376210813834725647961191$
% GP-Test A357425(peak_s)== 183376210813834725647961191
for
 $s / \log n = 7.722 \MyTightDots$
% GP-Test  my(r=bound_of_s(peak_s)); print(r); \
% GP-Test  nearly_equal(r, 7.722, 1e-3)
 \kern.1em
so a factor at least that large is required.

\medskip

See \href{http://oeis.org/A363758}{A363758} for the largest sum $s$
possible within a given number of digits.

%------------------------------------------------------------------------------
\pagebreak

% GP-DEFINE  A364779 = OEIS_data_func("A364779",'bfile);
% GP-DEFINE  low_of_s(s) = s / log(A364779(s)); \\ not exact

A corresponding possible lower bound factor $L$ would be,
\begin{align}
      L                     \mskip2mu
        \log n              \mskip2mu
    <                       \mskip2mu
a(n)                        \kern2.5em
    \text{for $\thickmuskip=\thinmuskip % around "\ge" operator
               n \ge 1$}
\notag
\end{align}

Again one way to experiment with such a factor is to take a sum of
digits $s$ and find the largest $n$ where sum
 $\thickmuskip=\medmuskip % around "=" operator
  a(n) = s$
occurs.  This is
 $\thickmuskip=\medmuskip % around "=" operator
  n = \mathrm{A364779}(s)$
and the following plot is the factor $L$ needed for a given $s$.

\begin{center}
\begin{tikzpicture}
  [xscale=.075,yscale=7]
  \coordinate (y-min) at (0, 1.9);
  \coordinate (y-max) at (0, 2.9);

  % generated by gen-bound-data.pl lower
  % x start 2
  \draw[line cap=butt] (2,2.88539)--(3,2.88539)--(3,2.164043)--(4,2.164043)
    --(4,2.48534)--(5,2.48534)--(5,2.404492)--(6,2.404492)--(6,2.164043)
    --(7,2.164043)--(7,2.470693)--(8,2.470693)--(8,2.308312)--(9,2.308312)
    --(9,2.378316)--(10,2.378316)--(10,2.282049)--(11,2.282049)--(11,1.983706)
    --(12,1.983706)--(12,2.162522)--(13,2.162522)--(13,2.225783)--(14,2.225783)
    --(14,2.283393)--(15,2.283393)--(15,2.443043)--(16,2.443043)--(16,2.488444)
    --(17,2.488444)--(17,2.330562)--(18,2.330562)--(18,2.373813)--(19,2.373813)
    --(19,2.413946)--(20,2.413946)--(20,2.540504)--(21,2.540504)--(21,2.573366)
    --(22,2.573366)--(22,2.518267)--(23,2.518267)--(23,2.548756)--(24,2.548756)
    --(24,2.577371)--(25,2.577371)--(25,2.684658)--(26,2.684658)--(26,2.555272)
    --(27,2.555272)--(27,2.580581)--(28,2.580581)--(28,2.604537)--(29,2.604537)
    --(29,2.324342)--(30,2.324342)--(30,2.404491)--(31,2.404491)--(31,2.428641)
    --(32,2.428641)--(32,2.506984)--(33,2.506984)--(33,2.528343)--(34,2.528343)
    --(34,2.54878)--(35,2.54878)--(35,2.568355)--(36,2.568355)--(36,2.58712)
    --(37,2.58712)--(37,2.597837)--(38,2.597837)--(38,2.515613)--(39,2.515613)
    --(39,2.487082)--(40,2.487082)--(40,2.550853)--(41,2.550853)--(41,2.522085)
    --(42,2.522085)--(42,2.501206)--(43,2.501206)--(43,2.475922)--(44,2.475922)
    --(44,2.533502)--(45,2.533502)--(45,2.507993)--(46,2.507993)--(46,2.48407)
    --(47,2.48407)--(47,2.538072)--(48,2.538072)--(48,2.552421)--(49,2.552421)
    --(49,2.605596)--(50,2.605596)--(50,2.579841)--(51,2.579841)--(51,2.631438)
    --(52,2.631438)--(52,2.53266)--(53,2.53266)--(53,2.545696)--(54,2.545696)
    --(54,2.593728)--(55,2.593728)--(55,2.600843)--(56,2.600843)--(56,2.548908)
    --(57,2.548908)--(57,2.496361)--(58,2.496361)--(58,2.540157)--(59,2.540157)
    --(59,2.551802)--(60,2.551802)--(60,2.472021)--(61,2.472021)--(61,2.483782)
    --(62,2.483782)--(62,2.5245)--(63,2.5245)--(63,2.535517)--(64,2.535517)
    --(64,2.489299)--(65,2.489299)--(65,2.446083)--(66,2.446083)--(66,2.431077)
    --(67,2.431077)--(67,2.467912)--(68,2.467912)--(68,2.478483)--(69,2.478483)
    --(69,2.488834)--(70,2.488834)--(70,2.524904)--(71,2.524904)--(71,2.534673)
    --(72,2.534673)--(72,2.518639)--(73,2.518639)--(73,2.503238)--(74,2.503238)
    --(74,2.488432)--(75,2.488432)--(75,2.52206)--(76,2.52206)--(76,2.483608)
    --(77,2.483608)--(77,2.492851)--(78,2.492851)--(78,2.525226)--(79,2.525226)
    --(79,2.534)--(80,2.534)--(80,2.542614)--(81,2.542614)--(81,2.528164)
    --(82,2.528164)--(82,2.559375)--(83,2.559375)--(83,2.522634)--(84,2.522634)
    --(84,2.530898)--(85,2.530898)--(85,2.493204)--(86,2.493204)--(86,2.522535)
    --(87,2.522535)--(87,2.509515)--(88,2.509515)--(88,2.51747)--(89,2.51747)
    --(89,2.525295)--(90,2.525295)--(90,2.532993)--(91,2.532993)--(91,2.540567)
    --(92,2.540567)--(92,2.52788)--(93,2.52788)--(93,2.535316)--(94,2.535316)
    --(94,2.487253)--(95,2.487253)--(95,2.513713)--(96,2.513713)--(96,2.520983)
    --(97,2.520983)--(97,2.50933)--(98,2.50933)--(98,2.516471)--(99,2.516471)
    --(99,2.505136)--(100,2.505136)--(100,2.424542)--(101,2.424542)
    --(101,2.431825)--(102,2.431825)--(102,2.455903)--(103,2.455903)
    --(103,2.446094)--(104,2.446094)--(104,2.469842)--(105,2.469842)
    --(105,2.47667)--(106,2.47667)--(106,2.483406)--(107,2.483406)
    --(107,2.490052)--(108,2.490052)--(108,2.463838)--(109,2.463838)
    --(109,2.470438)--(110,2.470438)--(110,2.493103)--(111,2.493103)
    --(111,2.483383)--(112,2.483383)--(112,2.489731)--(113,2.489731)
    --(113,2.480238)--(114,2.480238)--(114,2.470982)--(115,2.470982)
    --(115,2.492658)--(116,2.492658)--(116,2.498752)--(117,2.498752)
    --(117,2.520293)--(118,2.520293)--(118,2.480352)--(119,2.480352)
    --(119,2.501372)--(120,2.501372)--(120,2.507231)--(121,2.507231)
    --(121,2.468768)--(122,2.468768)--(122,2.474646)--(123,2.474646)
    --(123,2.49493)--(124,2.49493)--(124,2.403034)--(125,2.403034)
    --(125,2.395701)--(126,2.395701)--(126,2.401625)--(127,2.401625)
    --(127,2.420685)--(128,2.420685)--(128,2.426441)--(129,2.426441)
    --(129,2.393193)--(130,2.393193)--(130,2.411744)--(131,2.411744)
    --(131,2.417395)--(132,2.417395)--(132,2.435848)--(133,2.435848)
    --(133,2.441341)--(134,2.441341)--(134,2.446776)--(135,2.446776)
    --(135,2.426792)--(136,2.426792)--(136,2.432191)--(137,2.432191)
    --(137,2.412834)--(138,2.412834)--(138,2.418193)--(139,2.418193)
    --(139,2.435717)--(140,2.435717)--(140,2.440935)--(141,2.440935)
    --(141,2.433953)--(142,2.433953)--(142,2.439103)--(143,2.439103)
    --(143,2.451843)--(144,2.451843)--(144,2.421218)--(145,2.421218)
    --(145,2.403159)--(146,2.403159)--(146,2.419732)--(147,2.419732)
    --(147,2.413293)--(148,2.413293)--(148,2.42971)--(149,2.42971)
    --(149,2.423238)--(150,2.423238)--(150,2.428141)--(151,2.428141);
  \coordinate (x-min) at (0,0 |- y-min);
  \coordinate (x-max) at (151,0 |- y-min);
  % end generated

  % X axis
  \draw[densely dashed,->] (x-min) -- ($(x-max) + (0, 0)$)
    node[at end,above left,yshift=.5ex] {$s$};
  \coordinate (x-tick) at (0,-.012);
  \foreach \x in {3, 20, 40, ...,140} {
    \draw ($(\x,0) + (y-min)$) -- +(x-tick)
      +(.5,0) node[at end,below] {\x};
  }

  % Y axis
  \draw[->] (y-min) -- ($(y-max) + (0,.15)$)
    node[at end,below right,xshift=2em,yshift=-1ex]
        {$L = \dfrac{s}{\log \mathrm{A364779}(s)}$
         \kern.4em bound factor};
  \coordinate (y-tick) at (-2,0);
  \foreach \y in {2.0, 2.5, 3.0} {
    \draw (0,\y) -- +(y-tick)
      node[at end,left=.1em,inner xsep=0pt] {\y};
  }

  % s=11 low
  \coordinate (low) at (11.5, 1.965);
  \draw[densely dashed] ($(low) +(10,.01)$)
    .. controls +(-170:.05) and +(-70:.045) .. (low)
    node[at start,above right,align=left,
         inner ysep=0pt,yshift=-1ex]
        {low at $s {=} 11$ \\
         $1.9837 \MyTightDots$};
% GP-DEFINE  low_s = 11;
% GP-Test  my(r=low_of_s(low_s)); print(r); \
% GP-Test  nearly_equal(r, 1.9837, 1e-4)
% for(s=3,12, print(low_of_s(s)))

% GP-Test  low_s == 11
% GP-Test  my(pos, v=vector(135,s, if(s<3,oo, low_of_s(s)))); \
% GP-Test    vecmin(v,&pos); pos == low_s

  % s=98 drop
  \coordinate (drop) at (124.5, 2.37);
  \draw[densely dashed] ($(drop) +(-1,-.05)$)
    .. controls +(80:.02) and +(-90:.05) .. (drop)
    node[at start,below] {$s {=} 124$};
% GP-DEFINE  drop_s = 124;
% GP-Test  my(r=low_of_s(drop_s)); print(r); \
% GP-Test  nearly_equal(r, 2.403, 1e-4)
% for(s=110,130, print(s"  "low_of_s(s)))

\end{tikzpicture}
\end{center}

The low (so far) at $s {=} 11$ might be an initial exception, but it's
not particularly obvious whether the rest might be working its way
down, or converging on something.

Drops such as at $s{=}124$
% GP-Test  drop_s == 124
are where $\mathrm{A364779}(s)$ has a relatively large increase.

\medskip

See \href{http://oeis.org/A364751}{A364751} for the smallest sum $s$
possible within a given number of digits.



%------------------------------------------------------------------------------
\end{document}
