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 A236360 Numerator of the mean of all parts of all partitions of n. 2
 1, 4, 3, 5, 7, 66, 35, 88, 135, 35, 56, 44, 1313, 63, 220, 48, 1683, 3465, 4655, 1254, 4158, 7348, 28865, 2700, 48950, 10556, 13545, 14872, 132385, 168120, 212102, 89056, 111573, 209270, 520905, 323586, 800569, 988570, 1216215, 35560, 1827903, 744436 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS The arithmetic mean, M(n), of all parts of all partitions of n can be approximated by n^e(n), as typified by these pairs: n ..... 100 .... 1000 .... 2000 .... 3000 .... 4000 .... 5000 e(n) .. 0.331 .. 0.3410 .. 0.3447 .. 0.3468 .. 0.3483 .. 0.3495 LINKS Alois P. Heinz, Table of n, a(n) for n = 1..1000 FORMULA M(n) = A066186(n)/A006128(n). EXAMPLE First eight means:  1, 4/3, 3/2, 5/3, 7/4, 66/35, 35/18, 88/43. MAPLE b:= proc(n, i) option remember; `if`(n=0, [1, 0\$2],       `if`(i<1, [0\$3], b(n, i-1)+`if`(i>n, [0\$3],        (l-> l+[0, l[1]*i, l[1]])(b(n-i, i)))))     end: a:= n-> numer((l->l[2]/l[3])(b(n\$2))): seq(a(n), n=1..50);  # Alois P. Heinz, Feb 06 2014 MATHEMATICA f[n_] := Sum[DivisorSigma[0, m] PartitionsP[n - m], {m, 1, n}]; u =  PartitionsP[Range[50]] Range[50]; t = Table[u[[n]]/f[n], {n, 1, 50}] Numerator[t]    (*A236360*) Denominator[t]  (*A234361*) means = Map[Mean[Flatten[IntegerPartitions[#]]] &, Range[50]]; pwrLaw = a x^b; fit = FindFit[means, pwrLaw, {a, b}, x]; Show[{ListPlot[means], Plot[Function[{x}, Evaluate[pwrLaw /. fit]][x], {x, 1, Length[means]}]}] fit  (* Peter J. C. Moses, Jan 22 2014 *) CROSSREFS Cf. A006128, A066186, A236361. Sequence in context: A257120 A306835 A033546 * A299323 A010475 A256367 Adjacent sequences:  A236357 A236358 A236359 * A236361 A236362 A236363 KEYWORD nonn,frac,easy AUTHOR Clark Kimberling, Jan 24 2014 STATUS approved

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Last modified October 29 18:53 EDT 2020. Contains 338067 sequences. (Running on oeis4.)