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

1,1

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.

Links

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

Example

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.

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 *)

Prog

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

Crossrefs

Equals A020492 \ A003601.

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

Cf. A342103, A342105, A342106.

Sequence in context: A265092 A262032 A074200 * A378118 A274223 A280734

Adjacent sequences: A342101 A342102 A342103 * A342105 A342106 A342107

Keyword

nonn

Author

Bernard Schott, Feb 28 2021

Extensions

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

Status

approved