A066571 -id:A066571 - cp's OEIS frontend

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

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 *)

A066572 Number of sets of distinct positive integers with geometric mean n.

Original entry on oeis.org

1, 3, 3, 9, 3, 255, 3, 31, 9, 255, 3, 48891, 3, 255, 255, 117, 3, 48891, 3, 48891, 255, 255, 3, 12896331, 9, 255, 31, 48891, 3, 329166915027, 3, 479, 255, 255, 255, 668187863, 3, 255, 255, 12896331, 3, 329166915027, 3, 48891, 48891, 255, 3, 3981060137, 9, 48891, 255, 48891, 3, 12896331, 255, 12896331, 255, 255, 3

Offset: 1

Amarnath Murthy, Dec 19 2001

a(m) = a(n) if m and n have the same factorization structure.
a(60) is approximately 9.3492e20. - Franklin T. Adams-Watters, Jun 09 2006
Observe that for any prime p, a(p^k) = A066571(k+1) and the largest set is the powers 0..2k of p.

			a(2) = 3 as there are three sets viz. {2}, {1,4}, {1,2,4}, each of which has geometric mean 2.
a(4) = 9: the nine sets are {4}, {1, 16}, {2, 8}, {1, 4, 16}, {2, 4, 8}, {1, 2, 32}, {1, 2, 4, 32}, {1, 2, 8, 16}, {1, 2, 4, 8, 16}.

Martin Fuller, Table of n, a(n) for n = 1..359
Martin Fuller, Python program

Cf. A066571.

(* Recomputation using existing values and prime signatures *)
a[1] = 1; a[n_] := Switch[ FactorInteger[n][[All, 2]] // Sort, {1}, 3, {2}, 9, {3}, 31, {4}, 117, {1, 1}, 255, {5}, 479, {1, 2}, 48891, {1, 3}, 12896331, {2, 2}, 668187863, {1, 4}, 3981060137, {1, 1, 1}, 329166915027, , 0]; Table[ a[n], {n, 1, 59}] (* _Jean-François Alcover, Sep 04 2013 *)

More terms from Naohiro Nomoto, Dec 26 2001
More terms from Franklin T. Adams-Watters, Jun 09 2006
More terms from Jean-François Alcover, Sep 04 2013

cp's OEIS Frontend

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

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