cp's OEIS Frontend

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

%I A342105 #16 Dec 01 2024 03:41:07
%S A342105 5,7,11,13,17,19,20,21,22,23,27,29,31,33,37,38,39,41,43,44,45,46,47,
%T A342105 49,51,53,54,55,57,59,60,61,62,65,66,67,68,69,71,73,77,79,83,85,86,87,
%U A342105 89,91,92,93,94,95,96,97,99,101,102,103,107,109,110,111,113,114,115,116
%N A342105 Arithmetic numbers (A003601) that are not balanced numbers (A020492).
%C A342105 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).
%C A342105 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.
%H A342105 Amiram Eldar, <a href="/A342105/b342105.txt">Table of n, a(n) for n = 1..10000</a>
%e A342105 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.
%p A342105 with(numtheory): filter:= q -> (sigma(q) mod tau(q) = 0) and (sigma(q) mod phi(q) <> 0) : select(filter, [$1..120]);
%t A342105 Select[Range[120], Divisible[DivisorSigma[1, #], {DivisorSigma[0, #], EulerPhi[#]}] == {True, False} &] (* _Amiram Eldar_, Mar 05 2021 *)
%o A342105 (PARI) isok(m) = my(s=sigma(m)); !(s % numdiv(m)) && (s % eulerphi(m)); \\ _Michel Marcus_, Mar 05 2021
%Y A342105 Equals A003601 \ A020492.
%Y A342105 Cf. A000005 (tau), A000010 (phi), A000203 (sigma), A102187 (sigma/tau).
%Y A342105 Cf. A342103, A342104, A342106.
%K A342105 nonn
%O A342105 1,1
%A A342105 _Bernard Schott_, Mar 05 2021