A066572 -id:A066572 - cp's OEIS frontend

A066571 Number of sets of positive integers with arithmetic mean n.

1, 3, 9, 31, 117, 479, 2061, 9183, 42021, 196239, 931457, 4480531, 21793257, 107004891, 529656765, 2640160039, 13241371629, 66771501151, 338333343825, 1721768732423, 8796192611917, 45096680384635, 231945566136129, 1196461977291959, 6188390166782849

Offset: 1

Amarnath Murthy, Dec 19 2001

From Franklin T. Adams-Watters, Sep 13 2011: (Start)
If we use nonnegative integers instead of positive integers, we get this sequence shifted left (i.e., with offset 0).
The largest number that can be included in set of positive integers with mean n is the triangular number n*(n+1)/2 = A000217(n).
All values are odd. Sets including n are paired with the same set with n removed, with exception of {n}, as the empty set has no average.
(End)

			a(2) = 3 as there are three sets viz. {2}, {1,3}, {1,2,3}, each of which has the arithmetic mean 2.
a(3) = 9: the nine sets are {3}, {1, 5}, {2, 4}, {1, 2, 6}, {1, 3, 5}, {2, 3, 4}, {1, 2, 3, 6}, {1, 2, 4, 5}, {1, 2, 3, 4, 5}.

Martin Fuller, Table of n, a(n) for n = 1..200

Cf. A066572, A000217.

Cf. A072701, A164283.

a066571 n = f [1..] 1 n 0 where
   f (k:ks) l nl x
     | y > nl  = 0
     | y < nl  = f ks (l + 1) (nl + n) y + f ks l nl x
     | otherwise = if y `mod` l == 0 then 1 else 0
     where y = x + k
-- Reinhard Zumkeller, Feb 13 2013

g := k->expand(mul(1+t*x^i,i=1..k)); A066571 := proc(n) local k; add(coeff(coeff(g(n*k),t,k),x,n*k),k=1..2*n-1); end;

g[k_] := Expand[Product[1 + t*x^i, {i, 1, k}]]; a[n_] := Sum[Coefficient[ Coefficient[g[n*k], t, k], x, n*k], {k, 1, 2*n - 1}]; Table[an = a[n]; Print["a(", n, ") = ", an]; an, {n, 1, 10}] (* Jean-François Alcover, Feb 10 2018, translated from Maple *)

Sum of coefficient of t^k x^(n*k) in Product_{i=1..n*k} (1+t*x^i) for k <= 2*n-1. - N. J. A. Sloane
From Martin Fuller, Sep 14 2023: (Start)
Constant term in formal Laurent series (Product_{i=1-n..n*(n-1)/2} (1+x^i)) - 1.
a(n) = (Sum_{i=0..n*(n-1)/2} A053632(n-1,i)*A000009(i))*2-1. (End)

Corrected and extended by N. J. A. Sloane, Dec 19 2001
More terms from Naohiro Nomoto, Jun 19 2002
More terms from David Wasserman, Sep 10 2002
More terms from Martin Fuller, Sep 14 2023

A164283 Number of ways to write n as the root-mean-square (RMS) of a set of distinct positive integers.

Original entry on oeis.org

1, 1, 1, 1, 3, 9, 19, 79, 225, 693, 1901, 5597, 17641, 57503, 195431, 647139, 2182987, 7344451, 25057681, 85742999, 295284367, 1028155825, 3596134963, 12659796475, 44696280143, 158226554179, 562623263251, 2006471222195, 7182910999719, 25795458946677, 92875047372825, 335362896810137

Offset: 1

Alois P. Heinz, Aug 12 2009

nonn

			a(6) = 9, because 6 is the RMS of 9 sets of distinct positive integers: 6 = RMS(6) = RMS(1,3,5,8,9) = RMS(3,4,5,7,9) = RMS(1,2,4,5,7,11) = RMS(1,3,5,6,8,9) = RMS(3,4,5,6,7,9) = RMS(1,2,3,5,7,8,10) = RMS(1,2,4,5,6,7,11) = RMS(1,2,3,5,6,7,8,10).

Eric Weisstein's World of Math, Root-Mean-Square

Cf. A163974, A066572, A066571, A072701.

Cf. A000196, A211868.

a164283 n = f [1..] 1 nn 0 where
   f (k:ks) l nl xx
     | yy > nl  = 0
     | yy < nl  = f ks (l + 1) (nl + nn) yy + f ks l nl xx
     | otherwise = if w == n then 1 else 0
     where w = if r == 0 then a000196 m else 0
           (m, r) = divMod yy l
           yy = xx + k * k
   nn = n ^ 2
-- Reinhard Zumkeller, Feb 13 2013

sns:= proc(i) option remember; `if`(i=1, 1, sns(i-1) +i^2) end: b:= proc(n, i, t) if n<0 or i

sns[i_] := sns[i] = If[i == 1, 1, sns[i-1] + i^2] ; b[n_, i_, t_] := Which[n < 0 || i < t, 0, n == 0, If[t == 0, 1, 0], i == 1, If[n == 1 && t == 1, 1, 0], True, b[n, i, t] = b[n, i-1, t] + b[n - i^2, i-1, t-1]]; a[n_] := a[n] = Module[{s = 1, k}, For[k = 2, sns[k] <= k*n^2, k++, s = s + b[k*n^2, Floor[Sqrt[k*n^2 - sns[k-1]]], k]]; s]; Table[Print[an = a[n]]; an, {n, 1, 29}] (* Jean-François Alcover, Dec 30 2013, translated from Maple *)

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A066571 Number of sets of positive integers with arithmetic mean n.

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