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

1,1

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.

Links

Table of n, a(n) for n=1..66.

Example

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.

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 *)

Prog

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

Crossrefs

Equals A003601 \ A020492.

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

Cf. A342103, A342104, A342106.

Sequence in context: A287517 A175100 A270998 * A035035 A256387 A113909

Adjacent sequences: A342102 A342103 A342104 * A342106 A342107 A342108

Keyword

nonn

Author

Bernard Schott, Mar 05 2021

Status

approved