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

1,2

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.

Links

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

Example

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.

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 *)

Crossrefs

Cf. A020492, A291566.

Sequence in context: A102150 A039588 A024579 * A299545 A068603 A045878

Adjacent sequences: A291562 A291563 A291564 * A291566 A291567 A291568

Keyword

nonn

Author

Jud McCranie, Aug 26 2017

Status

approved