cp's OEIS Frontend

A342103 Balanced numbers (A020492) that are also arithmetic numbers (A003601).

%I A342103 #21 Dec 01 2024 03:41:14
%S A342103 1,3,6,14,15,30,35,42,56,70,78,105,140,168,190,210,248,264,270,357,
%T A342103 418,420,570,594,616,630,714,744,812,840,910,1045,1240,1254,1485,1672,
%U A342103 1848,2090,2214,2376,2436,2580,2730,2970,3080,3135,3339,3596,3720,3828,3956,4064,4180
%N A342103 Balanced numbers (A020492) that are also arithmetic numbers (A003601).
%C A342103 Equivalently, numbers m such that phi(m) (A000010) and tau(m) (A000005) both divide sigma(m) (A000203). In this case, the quotients sigma(m)/phi(m) = A023897(m) and sigma(m)/tau(m) = A102187(m).
%C A342103 Phi, tau and sigma are multiplicative functions and for this reason if k and q are coprime and included in this sequence then k*q is another term.
%C A342103 The only prime in the sequence is 3, because sigma(2)/tau(2) = 3/2 and when p is an odd prime, sigma(p)/phi(p) = (p+1)/(p-1) is an integer iff p=3 with sigma(3)/phi(3) = 4/2 = 2, and also sigma(3)/tau(3) = 4/2 = 2.
%H A342103 Amiram Eldar, <a href="/A342103/b342103.txt">Table of n, a(n) for n = 1..10000</a>
%e A342103 phi(30) = tau(30) = 8, sigma(30) = 72 and 72/8 = 9, hence 30 is a term.
%e A342103 phi(12) = 4, tau(12) = 6, sigma(12) = 28, phi(12) divides sigma(12), but tau(12) does not divide sigma(12), hence 12 is a balanced number but is not an arithmetic number, and 12 is not a term.
%e A342103 phi(20) = 8, tau(20) = 6, sigma(20) = 42, tau(20) divides sigma(20), but phi(20) does not divide sigma(20), hence 20 is an arithmetic number but is not a balanced number, and 20 is not a term.
%p A342103 with(numtheory): filter:= q -> (sigma(q) mod phi(q) = 0) and (sigma(q) mod tau(q) = 0) : select(filter, [$1..5000]);
%t A342103 Select[Range[5000], And @@ Divisible[DivisorSigma[1, #], {DivisorSigma[0, #], EulerPhi[#]}] &] (* _Amiram Eldar_, Feb 28 2021 *)
%o A342103 (PARI) isok(m) = my(s=sigma(m)); !(s % eulerphi(m)) && !(s % numdiv(m)); \\ _Michel Marcus_, Mar 01 2021
%Y A342103 Intersection of A003601 and A020492.
%Y A342103 Cf. A000005 (tau), A000010 (phi), A000203 (sigma), A023897 (sigma/phi), A102187 (sigma/tau).
%Y A342103 Cf. A342104, A342105, A342106.
%K A342103 nonn
%O A342103 1,2
%A A342103 _Bernard Schott_, Feb 28 2021