A066572 - OEIS

%I #23 Aug 18 2024 21:55:36

%S 1,3,3,9,3,255,3,31,9,255,3,48891,3,255,255,117,3,48891,3,48891,255,

%T 255,3,12896331,9,255,31,48891,3,329166915027,3,479,255,255,255,

%U 668187863,3,255,255,12896331,3,329166915027,3,48891,48891,255,3,3981060137,9,48891,255,48891,3,12896331,255,12896331,255,255,3

%N Number of sets of distinct positive integers with geometric mean n.

%C a(m) = a(n) if m and n have the same factorization structure.

%C a(60) is approximately 9.3492e20. - _Franklin T. Adams-Watters_, Jun 09 2006

%C Observe that for any prime p, a(p^k) = A066571(k+1) and the largest set is the powers 0..2k of p.

%H Martin Fuller, <a href="/A066572/b066572.txt">Table of n, a(n) for n = 1..359</a>

%H Martin Fuller, <a href="/A066572/a066572.txt">Python program</a>

%e a(2) = 3 as there are three sets viz. {2}, {1,4}, {1,2,4}, each of which has geometric mean 2.

%e 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}.

%t (* Recomputation using existing values and prime signatures *)

%t 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 *)

%Y Cf. A066571.

%K nonn,nice

%O 1,2

%A _Amarnath Murthy_, Dec 19 2001

%E More terms from _Naohiro Nomoto_, Dec 26 2001

%E More terms from _Franklin T. Adams-Watters_, Jun 09 2006

%E More terms from _Jean-François Alcover_, Sep 04 2013