A342105 Arithmetic numbers (A003601) that are not balanced numbers (A020492).

5, 7, 11, 13, 17, 19, 20, 21, 22, 23, 27, 29, 31, 33, 37, 38, 39, 41, 43, 44, 45, 46, 47, 49, 51, 53, 54, 55, 57, 59, 60, 61, 62, 65, 66, 67, 68, 69, 71, 73, 77, 79, 83, 85, 86, 87, 89, 91, 92, 93, 94, 95, 96, 97, 99, 101, 102, 103, 107, 109, 110, 111, 113, 114, 115, 116

Offset: 1

Author

Bernard Schott, Mar 05 2021

Keywords

• nonn

Comments

Equivalently, numbers m such that tau(m) divides sigma(m) but phi(m) does not divide sigma(m), the corresponding quotients sigma(m)/tau(m) = A102187(m).

Primes in the sequence are primes >= 5; proof: 2 is in A342104 and 3 is in A342103, then for p prime >= 5, phi(p) = p-1 >= 4, tau(p) = 2, sigma(p) = p+1 >= 6; hence 2 divides p+1 but p-1 does not divide p+1.

Examples

			Sigma(21) = 32, tau(21) = 4 and phi(21) = 12, hence tau(21) divides sigma(21), but phi(21) does not divide sigma(21), so 21 is a term.
		

Links

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

Crossrefs

Equals A003601 \ A020492.

Cf. A000005 (tau), A000010 (phi), A000203 (sigma), A102187 (sigma/tau).

Cf. A342103, A342104, A342106.

Programs

Maple

with(numtheory): filter:= q -> (sigma(q) mod tau(q) = 0) and (sigma(q) mod phi(q) <> 0) : select(filter, [$1..120]);

Mathematica

Select[Range[120], Divisible[DivisorSigma[1, #], {DivisorSigma[0, #], EulerPhi[#]}] == {True, False} &] (* Amiram Eldar, Mar 05 2021 *)

PARI

isok(m) = my(s=sigma(m)); !(s % numdiv(m)) && (s % eulerphi(m)); \\ Michel Marcus, Mar 05 2021