A291566 Non-primitive balanced numbers: balanced numbers of the form m*n where m, n > 1 are both balanced.

6, 30, 42, 70, 105, 168, 210, 420, 570, 714, 744, 840, 1254, 1848, 2090, 2436, 2730, 2970, 3135, 3720, 5016, 6270, 6678, 8680, 9240, 10098, 10788, 11868, 12180, 12192, 12540, 13566, 14630, 15834, 16188, 20790, 21318, 24024, 24882, 25080, 25908, 26040, 26796, 32130, 43890, 48360

Offset: 1

Author

Jud McCranie, Aug 26 2017

Keywords

• nonn

Comments

A positive integer, n, is a balanced number (A020492) if sigma(n) is a multiple of phi(n). Since phi and sigma are multiplicative functions, if m and n are balanced numbers and gcd(m,n)=1, mn is also a balanced number. This sequence consists of only these imprimitive terms.

Examples

			2 and 3 are balanced numbers, gcd(2,3)=1, so 6 is a non-primitive balanced number.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..5000

Crossrefs

Cf. A020492, A291565.

Programs

Mathematica

balQ[n_] := Divisible[DivisorSigma[1,n], EulerPhi[n]]; nonprimQ[n_] := balQ[n] && Module[{d = Divisors[n], ans = False}, Do[If[GCD[d[[k]], n/d[[k]]]==1 && balQ[ d[[k]]] && balQ[n/d[[k]]], ans=True; Break[]], {k, 2, Floor[Length[d]/2]}]; ans]; Select[Range[50000], nonprimQ] (* Amiram Eldar, Jun 26 2019 *)