\documentclass[a4paper]{article} \pagestyle{empty} % no page numbers % Title and author go into PDF (by hyperref pdfusetitle), % but not actually displayed. \title{A363758 Maximum Sum of Digits} \author{Kevin Ryde} \date{March 2024} \usepackage[T1]{fontenc} % T1 for accents, before babel \usepackage{amsmath} \usepackage[pdfusetitle, pdfsubject={OEIS A363758}, pdfkeywords={OEIS, A363758, base 4/3, mean digit}, 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("OEIS-data.gp"); % GP-DEFINE read("OEIS-data-wip.gp"); % GP-DEFINE A363758 = OEIS_data_func("A363758",'bfile); % GP-DEFINE A363758_maxn = #OEIS_data("A363758",'bfile) - 1; % GP-DEFINE print("A363758_maxn = ",A363758_maxn); % GP-DEFINE mean(n) = A363758(n) / n; %------------------------------------------------------------------------------ \begin{document} \hfil {\large A363758 Maximum Sum of Digits} \hfil \vspace{.2ex} \hfil {Kevin Ryde, March 2024} \hfil \newcommand\MyTightDots{.\kern.08em.\kern.08em.} \vspace{\baselineskip} \href{http://oeis.org/A363758}{A363758} is the maximum sum of digits for an $n$ digit number in fractional base $4/3$. The following is a plot of $a(n) / n$ which is the mean digit in such a number. \begin{center} \begin{tikzpicture} [xscale=.0554,yscale=20, my x number/.style={inner xsep=.1em}, ] \coordinate (y-min) at (0, 2.0); \coordinate (y-max) at (0, 2.43); % generated by gen-plot-mean.pl % skip initial terms 0,3/1,6/2,8/3 \draw[line cap=butt] (4,0 |- y-max)--(4,9/4)--(5,9/4)--(5,12/5)--(6,12/5) --(6,13/6)--(7,13/6)--(7,15/7)--(8,15/7)--(8,17/8)--(9,17/8)--(9,19/9) --(10,19/9)--(10,22/10)--(11,22/10)--(11,24/11)--(12,24/11)--(12,26/12) --(13,26/12)--(13,28/13)--(14,28/13)--(14,30/14)--(15,30/14)--(15,32/15) --(16,32/15)--(16,33/16)--(17,33/16)--(17,36/17)--(18,36/17)--(18,37/18) --(19,37/18)--(19,40/19)--(20,40/19)--(20,42/20)--(21,42/20)--(21,44/21) --(22,44/21)--(22,46/22)--(23,46/22)--(23,48/23)--(24,48/23)--(24,50/24) --(25,50/24)--(25,52/25)--(26,52/25)--(26,54/26)--(27,54/26)--(27,56/27) --(28,56/27)--(28,57/28)--(29,57/28)--(29,60/29)--(30,60/29)--(30,62/30) --(31,62/30)--(31,65/31)--(32,65/31)--(32,67/32)--(33,67/32)--(33,70/33) --(34,70/33)--(34,71/34)--(35,71/34)--(35,73/35)--(36,73/35)--(36,75/36) --(37,75/36)--(37,77/37)--(38,77/37)--(38,80/38)--(39,80/38)--(39,83/39) --(40,83/39)--(40,84/40)--(41,84/40)--(41,87/41)--(42,87/41)--(42,90/42) --(43,90/42)--(43,93/43)--(44,93/43)--(44,94/44)--(45,94/44)--(45,96/45) --(46,96/45)--(46,98/46)--(47,98/46)--(47,101/47)--(48,101/47)--(48,104/48) --(49,104/48)--(49,106/49)--(50,106/49)--(50,108/50)--(51,108/50) --(51,109/51)--(52,109/51)--(52,112/52)--(53,112/52)--(53,115/53) --(54,115/53)--(54,117/54)--(55,117/54)--(55,120/55)--(56,120/55) --(56,122/56)--(57,122/56)--(57,123/57)--(58,123/57)--(58,126/58) --(59,126/58)--(59,129/59)--(60,129/59)--(60,131/60)--(61,131/60) --(61,133/61)--(62,133/61)--(62,134/62)--(63,134/62)--(63,137/63) --(64,137/63)--(64,139/64)--(65,139/64)--(65,141/65)--(66,141/65) --(66,144/66)--(67,144/66)--(67,146/67)--(68,146/67)--(68,149/68) --(69,149/68)--(69,151/69)--(70,151/69)--(70,153/70)--(71,153/70) --(71,156/71)--(72,156/71)--(72,158/72)--(73,158/72)--(73,161/73) --(74,161/73)--(74,164/74)--(75,164/74)--(75,165/75)--(76,165/75) --(76,168/76)--(77,168/76)--(77,171/77)--(78,171/77)--(78,173/78) --(79,173/78)--(79,175/79)--(80,175/79)--(80,177/80)--(81,177/80) --(81,180/81)--(82,180/81)--(82,181/82)--(83,181/82)--(83,184/83) --(84,184/83)--(84,187/84)--(85,187/84)--(85,190/85)--(86,190/85) --(86,191/86)--(87,191/86)--(87,193/87)--(88,193/87)--(88,195/88) --(89,195/88)--(89,197/89)--(90,197/89)--(90,200/90)--(91,200/90) --(91,203/91)--(92,203/91)--(92,204/92)--(93,204/92)--(93,207/93) --(94,207/93)--(94,210/94)--(95,210/94)--(95,213/95)--(96,213/95) --(96,214/96)--(97,214/96)--(97,215/97)--(98,215/97)--(98,218/98) --(99,218/98)--(99,220/99)--(100,220/99)--(100,222/100)--(101,222/100) --(101,225/101)--(102,225/101)--(102,227/102)--(103,227/102)--(103,229/103) --(104,229/103)--(104,232/104)--(105,232/104)--(105,235/105)--(106,235/105) --(106,238/106)--(107,238/106)--(107,241/107)--(108,241/107)--(108,242/108) --(109,242/108)--(109,245/109)--(110,245/109)--(110,248/110)--(111,248/110) --(111,251/111)--(112,251/111)--(112,254/112)--(113,254/112)--(113,256/113) --(114,256/113)--(114,259/114)--(115,259/114)--(115,262/115)--(116,262/115) --(116,265/116)--(117,265/116)--(117,268/117)--(118,268/117)--(118,271/118) --(119,271/118)--(119,274/119)--(120,274/119)--(120,277/120)--(121,277/120) --(121,279/121)--(122,279/121)--(122,280/122)--(123,280/122)--(123,281/123) --(124,281/123)--(124,284/124)--(125,284/124)--(125,285/125)--(126,285/125) --(126,288/126)--(127,288/126)--(127,291/127)--(128,291/127)--(128,292/128) --(129,292/128)--(129,294/129)--(130,294/129)--(130,297/130)--(131,297/130) --(131,300/131)--(132,300/131)--(132,303/132)--(133,303/132)--(133,304/133) --(134,304/133)--(134,307/134)--(135,307/134)--(135,309/135)--(136,309/135) --(136,312/136)--(137,312/136)--(137,313/137)--(138,313/137)--(138,315/138) --(139,315/138)--(139,316/139)--(140,316/139)--(140,319/140)--(141,319/140) --(141,321/141)--(142,321/141)--(142,324/142)--(143,324/142)--(143,326/143) --(144,326/143)--(144,329/144)--(145,329/144)--(145,331/145)--(146,331/145) --(146,334/146)--(147,334/146)--(147,337/147)--(148,337/147)--(148,339/148) --(149,339/148)--(149,342/149)--(150,342/149)--(150,343/150)--(151,343/150) --(151,345/151)--(152,345/151)--(152,347/152)--(153,347/152)--(153,350/153) --(154,350/153)--(154,352/154)--(155,352/154)--(155,354/155)--(156,354/155) --(156,356/156)--(157,356/156)--(157,359/157)--(158,359/157)--(158,361/158) --(159,361/158)--(159,363/159)--(160,363/159)--(160,366/160)--(161,366/160) --(161,369/161)--(162,369/161)--(162,372/162)--(163,372/162)--(163,373/163) --(164,373/163)--(164,374/164)--(165,374/164)--(165,376/165)--(166,376/165) --(166,379/166)--(167,379/166)--(167,380/167)--(168,380/167)--(168,383/168) --(169,383/168)--(169,384/169)--(170,384/169)--(170,387/170)--(171,387/170) --(171,389/171)--(172,389/171)--(172,392/172)--(173,392/172)--(173,394/173) --(174,394/173)--(174,397/174)--(175,397/174)--(175,399/175)--(176,399/175) --(176,400/176)--(177,400/176)--(177,402/177)--(178,402/177)--(178,404/178) --(179,404/178)--(179,407/179)--(180,407/179)--(180,409/180)--(181,409/180) --(181,411/181)--(182,411/181)--(182,414/182)--(183,414/182)--(183,417/183) --(184,417/183)--(184,419/184)--(185,419/184)--(185,421/185)--(186,421/185) --(186,424/186)--(187,424/186)--(187,427/187)--(188,427/187)--(188,429/188) --(189,429/188)--(189,430/189)--(190,430/189)--(190,432/190)--(191,432/190) --(191,434/191)--(192,434/191)--(192,436/192)--(193,436/192)--(193,439/193) --(194,439/193)--(194,442/194)--(195,442/194)--(195,445/195)--(196,445/195) --(196,447/196)--(197,447/196)--(197,449/197)--(198,449/197)--(198,452/198) --(199,452/198)--(199,453/199)--(200,453/199)--(200,456/200)--(201,456/200) --(201,459/201)--(202,459/201)--(202,462/202)--(203,462/202)--(203,464/203) --(204,464/203)--(204,467/204)--(205,467/204)--(205,468/205)--(206,468/205) --(206,469/206)--(207,469/206)--(207,472/207)--(208,472/207); \coordinate (x-min) at (0,0 |- y-min); \coordinate (x-max) at (208,0 |- y-min); % end generated % X axis \draw[dashed,->] (x-min) -- ($(x-max) + (0, 0)$) node[at end,above left,inner xsep=0pt, yshift=.8ex] {$n$}; \coordinate (x-tick) at (0,-.012); \foreach \x in {4, 20, 40, ...,200} { \draw ($(\x,0) + (y-min)$) -- +(x-tick) +(.5,0) node[at end,below] {\x}; } % Y axis \draw[->] (y-min) -- ($(y-max) + (0,.02)$) node[at end,below right,xshift=3em,yshift=-2ex] {$\dfrac{a(n)}{n}$ \hskip.2em mean digit}; \coordinate (y-tick) at (-2,0); \foreach \y in {2.0, 2.1, 2.2, 2.3, 2.4} { \draw (0,\y) -- +(y-tick) node[at end,left=.1em,inner xsep=0pt] {\y}; } \coordinate (peak) at (120.5, 2.315); \draw[dashed] ($(peak) +(-5,.023)$) .. controls +(-45:.01) and +(135:.01) .. (peak) node[at start,above,align=left] {$n{=}120$ \\ $2.308 \MyTightDots$}; \end{tikzpicture} \end{center} \smallskip Initial terms $a(1 \MyTightDots 3)$ are omitted. Their means are $3, \, 3, \, 2{+}\tfrac23$. % GP-Test apply(mean,[1..3]) == [3, 3, 2+2/3] The peak (so far) in the middle of the plot is at $n {=} % GP-CONSTANT peak_n 120 % GP-END $ which has sum of digits $a(n) = 277$ % GP-Test A363758(peak_n) == 277 for mean $277/120 = 2.308 \MyTightDots$. % GP-Test mean(peak_n) == 277/120 % GP-Test floor(mean(peak_n)*1000) == 2308 % GP-Test my(m=vecmax(apply(mean,[6..A363758_maxn]))); \ % GP-Test m == 277/120 && \ % GP-Test select(n->mean(n)==m, [4..A363758_maxn]) == \ % GP-Test [peak_n] \bigskip If digits were random $0,1,2,3$ then the mean would be $1.5$. Some experiments with small $n$ suggest this is roughly so taken over all numbers of $n$ digits. The numerical maximum number with $n$ digits uses only $1,2,3$ (and ends $3$). If those were random then their mean would be $2$. Some experiments suggest this is roughly so for small $n$. The plot up to the middle peak $n{=}120$ might have looked like continuing to grow, but beyond that the means drift down (so far) towards about $2.28$. % end at n=200 a(n)=456 is mean 456/200 = 2.28 % GP-Test mean(200) == 456/200 % GP-Test mean(200) == 228/100 % % n=207 very close to 2.28 % GP-Test mean(207) == 472/207 % GP-Test mean(207) == 228/100 + 1/5175 % % for(n=A363758_maxn-20,A363758_maxn, \ % print(mean(n)*1.0)); %------------------------------------------------------------------------------ \end{document}