cp's OEIS Frontend

A341938 Numbers m such that the geometric mean of tau(m) and phi(m) is an integer where phi is the Euler totient function (A000010) and tau is the number of divisors function (A000005).

%I A341938 #22 Feb 27 2021 21:56:34
%S A341938 1,3,8,10,18,19,24,30,34,45,52,54,57,73,74,85,102,125,135,140,152,153,
%T A341938 156,163,182,185,190,202,219,222,252,255,333,342,360,375,394,416,420,
%U A341938 436,451,455,456,459,476,489,505,514,546,555,570,584,606,625,629,640,646,679,680,730
%N A341938 Numbers m such that the geometric mean of tau(m) and phi(m) is an integer where phi is the Euler totient function (A000010) and tau is the number of divisors function (A000005).
%C A341938 The first 11 terms of this sequence are also the first 11 terms of A341939: m such that phi(m)/tau(m) is the square of an integer. Indeed, if phi(m)/tau(m) is a perfect square then phi(m)*tau(m) is also a square, but the converse is false. These counterexamples are in A341940, the first one is a(12) = 54.
%C A341938 If k and q are terms and coprimes, then k*q is another term.
%C A341938 Some subsequences (see examples):
%C A341938 -> The seven terms that satisfy tau(m) = phi(m) form the subsequence A020488.
%C A341938 -> Primes p of the form 2*k^2 + 1 (A090698) form another subsequence because tau(p) = 2 and phi(p) = p-1 = 2*k^2, so tau(p)*phi(p) = (2*k)^2.
%C A341938 -> Cubes p^3 where p is a prime of the form k^2+1 (A002496) form another subset with tau(p^3)*phi(p^3) = (2*k*p)^2.
%e A341938 phi(18) = tau(18) = 6, so phi(18)*tau(18) = 6^2.
%e A341938 phi(19) = 18, tau(19) = 2, so phi(19)*tau(19) = 36 = 6^2.
%e A341938 phi(34) = 16, tau(34) = 4, so phi(34)*tau(34) = 16*4 = 64 = 8^2.
%e A341938 phi(125) = 100, tau(125) = 4, so phi(125)*tau(125) = 400 = 20^2.
%p A341938 with(numtheory): filter:= n -> issqr(phi(n)*tau(n)) : select(filter, [$1..750]);
%t A341938 Select[Range[1000], IntegerQ @ GeometricMean[{DivisorSigma[0, #], EulerPhi[#]}] &] (* _Amiram Eldar_, Feb 24 2021 *)
%o A341938 (PARI) isok(m) = issquare(numdiv(m)*eulerphi(m)); \\ _Michel Marcus_, Feb 24 2021
%Y A341938 Similar for: A011257 (phi*sigma square), A327830 (sigma*tau square).
%Y A341938 Subsequences: A020488, A090698.
%Y A341938 Cf. A341939, A341940.
%Y A341938 Cf. A000005 (tau), A000010 (phi).
%K A341938 nonn
%O A341938 1,2
%A A341938 _Bernard Schott_, Feb 24 2021