A291565 Primitive balanced numbers: primitive numbers not of the form m*n where m, n > 1 are both primitive.

1, 2, 3, 12, 14, 15, 35, 56, 78, 140, 190, 248, 264, 270, 357, 418, 594, 616, 630, 812, 910, 1045, 1240, 1485, 1672, 2214, 2376, 2580, 3080, 3339, 3596, 3828, 3956, 4064, 4180, 4522, 4674, 5049, 5278, 5396, 5544, 5940, 6426, 7110, 7668, 8008, 8636, 8932, 10659, 11160, 11880, 12441, 12648, 15642

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, if m and n are balanced numbers and gcd(m,n)=1, m*n is also a balanced number. This sequence eliminates these imprimitive terms.

Examples

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

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A020492, A291566.

Programs

Mathematica

balQ[n_] := Divisible[DivisorSigma[1,n], EulerPhi[n]]; primQ[n_] := balQ[n] && Module[{d = Divisors[n], ans = True}, Do[If[GCD[d[[k]], n/d[[k]]]==1 && balQ[ d[[k]]] && balQ[n/d[[k]]], ans=False; Break[]], {k, 2, Floor[Length[d]/2]}]; ans]; Select[Range[16000],primQ] (* Amiram Eldar, Jun 26 2019 *)