A342106 - OEIS

%I #13 Mar 08 2021 22:53:13

%S 4,8,9,10,16,18,24,25,26,28,32,34,36,40,48,50,52,58,63,64,72,74,75,76,

%T 80,81,82,84,88,90,98,100,104,106,108,112,117,120,121,122,124,128,130,

%U 136,144,146,148,152,156,160,162,170,171,172,175,176,178,180,192,194,196,200

%N Numbers that are not arithmetic and not balanced.

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

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

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

%e 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.

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

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

%o (PARI) isok(m) = my(s=sigma(m)); (s % numdiv(m)) && (s % eulerphi(m)); \\ _Michel Marcus_, Mar 08 2021

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

%Y Cf. A020492, A003601.

%Y Cf. A342103, A342104, A342105.

%K nonn

%O 1,1

%A _Bernard Schott_, Mar 08 2021