A342106 Numbers that are not arithmetic and not balanced.

4, 8, 9, 10, 16, 18, 24, 25, 26, 28, 32, 34, 36, 40, 48, 50, 52, 58, 63, 64, 72, 74, 75, 76, 80, 81, 82, 84, 88, 90, 98, 100, 104, 106, 108, 112, 117, 120, 121, 122, 124, 128, 130, 136, 144, 146, 148, 152, 156, 160, 162, 170, 171, 172, 175, 176, 178, 180, 192, 194, 196, 200

Offset: 1

Author

Bernard Schott, Mar 08 2021

Keywords

• nonn

Comments

Equivalently, numbers m such that neither tau(m) (A000005) nor phi(m) (A000010) divide sigma(m) (A000203).

A342103, A342104, A342105 and this sequence form a partition of the set of positive integers N* (A000027).

There are no primes in the sequence since 2 is in A342104, 3 is in A342103, and odd primes >= 5 are in A342105.

Examples

			Sigma(9) = 13, tau(9) = 3 and phi(9) = 6, hence sigma(9)/tau(9) = 13/3 and sigma(9)/phi(9) = 13/6, 9 is a term.
		

Crossrefs

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

Cf. A020492, A003601.

Cf. A342103, A342104, A342105.

Programs

Maple

with(numtheory): filter:= q -> (sigma(q) mod phi(q) <> 0) and (sigma(q) mod tau(q) <> 0) : select(filter, [$1..200]);

Mathematica

Select[Range[200], ! Or @@ Divisible[DivisorSigma[1, #], {DivisorSigma[0, #], EulerPhi[#]}] &] (* Amiram Eldar, Mar 08 2021 *)

PARI

isok(m) = my(s=sigma(m)); (s % numdiv(m)) && (s % eulerphi(m)); \\ Michel Marcus, Mar 08 2021