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

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

Author

Amarnath Murthy, Dec 19 2001

Keywords

• nonn
• nice

Comments

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.

Examples

			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}.
		

Links

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

Crossrefs

Cf. A066571.

Programs

Mathematica

(* 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 *)

Extensions

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