A342104 Balanced numbers (A020492) that are not arithmetic numbers (A003601).

2, 12, 18630, 27000, 443394, 6242022, 14412720, 22315419, 26744100, 44630838, 50496960, 106034880, 128710944, 148536990, 162907584, 212072880, 218470770, 296259930, 349444530, 397253968, 535267776, 641250900, 641418960, 666274653, 684165552, 688208724, 709639408

Offset: 1

Author

Bernard Schott, Feb 28 2021

Keywords

• nonn

Comments

Equivalently, numbers m such that phi(m) divides sigma(m) but tau(m) does not divide sigma(m), the corresponding quotients sigma(m)/phi(m) = A023897(m).

The only prime in the sequence is 2, because sigma(2)/phi(2) = 3 and sigma(2)/tau(2) = 3/2; then, if p odd prime, sigma(p)/phi(p) = (p+1)/(p-1) is an integer iff p = 3, but for p = 3, tau(3) divides sigma(3) with sigma(3)/tau(3) = 4/2 = 2.

Examples

			Sigma(12) = 28, phi(12) = 4 and tau(12) = 6, hence phi(12) divides sigma(12), but tau(12) does not divide sigma(12), so 12 is a term.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..5620

Crossrefs

Equals A020492 \ A003601.

Cf. A000005 (tau), A000010 (phi), A000203 (sigma), A023897 (sigma/phi).

Cf. A342103, A342105, A342106.

Programs

Maple

with(numtheory): filter:= q -> (sigma(q) mod phi(q) = 0) and (sigma(q) mod tau(q) <> 0) : select(filter, [$1..500000]);

Mathematica

Select[Range[500000], Divisible[DivisorSigma[1, #], {DivisorSigma[0, #], EulerPhi[#]}] == {False, True} &] (* Amiram Eldar, Feb 28 2021 *)

PARI

isok(m) = my(s=sigma(m)); !(s % eulerphi(m)) && (s % numdiv(m)); \\ Michel Marcus, Mar 01 2021

Extensions

a(5)-a(27) from Amiram Eldar, Feb 28 2021