\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 \begin{center} \begin{tikzpicture} [xscale=.0245,yscale=1.55] \coordinate (y-min) at (0, 2.0); \coordinate (y-max) at (0, 8.0); % generated by gen-bound-data.pl upper % x start 2 \draw[line cap=butt] (2,2.88539)--(3,2.88539)--(3,2.730718)--(4,2.730718) --(4,2.48534)--(5,2.48534)--(5,2.790553)--(6,2.790553)--(6,3.08339) --(7,3.08339)--(7,3.040061)--(8,3.040061)--(8,3.336259)--(9,3.336259) --(9,3.323424)--(10,3.323424)--(10,3.284587)--(11,3.284587)--(11,3.55867) --(12,3.55867)--(12,3.827148)--(13,3.827148)--(13,3.785687)--(14,3.785687) --(14,3.821418)--(15,3.821418)--(15,3.988089)--(16,3.988089)--(16,4.011046) 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--(324,7.598981)--(325,7.598981)--(325,7.57135)--(326,7.57135) --(326,7.594646)--(327,7.594646)--(327,7.567227)--(328,7.567227) --(328,7.590369)--(329,7.590369)--(329,7.61351)--(330,7.61351) --(330,7.586148)--(331,7.586148)--(331,7.609136)--(332,7.609136) --(332,7.581982)--(333,7.581982)--(333,7.604819)--(334,7.604819) --(334,7.627657)--(335,7.627657)--(335,7.600559)--(336,7.600559) --(336,7.623248)--(337,7.623248)--(337,7.645936)--(338,7.645936) --(338,7.618895)--(339,7.618895)--(339,7.641437)--(340,7.641437) --(340,7.614599)--(341,7.614599)--(341,7.636995)--(342,7.636995) --(342,7.659391)--(343,7.659391)--(343,7.632611)--(344,7.632611) --(344,7.606172)--(345,7.606172)--(345,7.628282)--(346,7.628282) --(346,7.602037)--(347,7.602037)--(347,7.624009)--(348,7.624009) --(348,7.597955)--(349,7.597955)--(349,7.619788)--(350,7.619788) --(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}