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Balanced numbers k (A020492) such that phi(k) is not divisible by 12.
1

%I #19 Nov 10 2024 09:25:05

%S 1,2,3,6,12,14,15,30,264,3828

%N Balanced numbers k (A020492) such that phi(k) is not divisible by 12.

%C The only balanced numbers k such that sigma(k) is not divisible by 12 are 1, 2, 3.

%C a(11) if it exists is greater than 10^8. - _Michel Marcus_, Aug 06 2013

%C a(11) if it exists is greater than 10^13. - _Giovanni Resta_, Jul 13 2015

%C a(11) if it exists is greater than 6.5*10^14 (checked using data from _Jud McCranie_). - _Amiram Eldar_, Nov 10 2024

%F k such that sigma(k)/phi(k) is an integer and phi(k) mod 12 != 0.

%p with(numtheory): BNM:=[]: for z from 1 to 1 do for n from 1 to 100000 do if phi(n) mod 12 > 0 and sigma(n) mod phi(n) = 0 then BNM:=[op(BNM),n] fi; od; od; BNM; # after 3828 there no others up to 2*10^6.

%t fQ[n_] := Block[{ds = DivisorSigma[1, n], ep = EulerPhi@n}, Mod[ep, 12] > 0 && IntegerQ[ds/ep]]; Do[ If[ fQ@n, Print@n], {n, 4*10^8}] (* _Robert G. Wilson v_, Jun 19 2006 *)

%Y Cf. A000010, A000203, A020492.

%K nonn,more,changed

%O 1,2

%A _Walter Kehowski_, Sep 13 2005