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A257639 a(n) is the minimal position at which the maximal value of row n appears in row n of triangle A008289.

%I A257639 #44 Jun 14 2018 08:30:25
%S A257639 1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,
%T A257639 5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,
%U A257639 6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7
%N A257639 a(n) is the minimal position at which the maximal value of row n appears in row n of triangle A008289.
%C A257639 Except rows 3, 4, 10, 11, 21 of triangle A008289, all the other rows up to row number 10^6 contain a single maximal value.
%C A257639 Conjecture: for n >= 22 there is a unique maximal value in row n of triangle A008289.
%H A257639 Gheorghe Coserea, <a href="/A257639/b257639.txt">Table of n, a(n) for n = 1..50000</a>
%H A257639 Paul Erdős, <a href="https://projecteuclid.org/euclid.bams/1183507709">On some asymptotic formulas in the theory of partitions</a>, Bull. Amer. Math. Soc. 52 (1946), no. 2, 185--188.
%H A257639 B. Richmond and A. Knopfmacher, <a href="http://gdz.sub.uni-goettingen.de/dms/load/img/?PPN=PPN356261603_0049&amp;IDDOC=178900">Compositions with distinct parts</a>, Aequationes Mathematicae 49 (1995), pp. 86-97.
%H A257639 G. Szekeres, <a href="http://qjmath.oxfordjournals.org/content/4/1/96.extract">Some asymptotic formulas in the theory of partitions (II)</a>, Quart. J. Math. Oxford (2), 4(1953), 96-111.
%F A257639 a(n) = min argmax(k->Q(n,k), k=1..m), that is a(n) = min{k, Q(n,k) = max{Q(n,p), p=1..m}}, where m = A003056(n) and Q(n,k) is defined by A008289.
%F A257639 a(n) ~ K*sqrt(n) + O(1), where K = 2*sqrt(3)*log(2)/Pi = 0.76430413884... (A131266).
%e A257639 For n=9, a(9)=2 because A003056(9)=3 and max{Q(9,p), p=1..3}=4 and Q(9,2)=4.
%o A257639 (PARI)
%o A257639 Q(N) = {
%o A257639   my(q = vector(N)); q[1] = [1, 0, 0, 0];
%o A257639   for (n = 2, N,
%o A257639     my(m = (sqrtint(8*n+1) - 1)\2);
%o A257639     q[n] = vector((1 + (m>>2)) << 2); q[n][1] = 1;
%o A257639     for (k = 2, m, q[n][k] = q[n-k][k] + q[n-k][k-1]));
%o A257639   return(q);
%o A257639 };
%o A257639 seq(N) = {
%o A257639   my(a = vector(N), q = Q(N), vmx = apply(vecmax, q));
%o A257639   for (n = 1, N, a[n] = vecmin(select(v->v==vmx[n], q[n], 1)));
%o A257639   a;
%o A257639 };
%o A257639 seq(86) \\ updated by _Gheorghe Coserea_, Jun 02 2018
%Y A257639 Cf. A003056, A008289, A030699, A131266.
%K A257639 nonn
%O A257639 1,5
%A A257639 _Gheorghe Coserea_, Nov 04 2015