cp's OEIS Frontend

A272798 Carmichael numbers k such that Euler totient function of k (phi(k)) is a perfect square.

%I A272798 #41 Apr 22 2024 10:01:41
%S A272798 1729,63973,75361,172081,278545,340561,658801,997633,1773289,3224065,
%T A272798 5310721,8719309,8719921,12945745,13187665,15888313,17586361,27402481,
%U A272798 29020321,39353665,40430401,49333201,67371265,84417985,120981601,128697361,129255841,130032865,151530401,151813201,158864833
%N A272798 Carmichael numbers k such that Euler totient function of k (phi(k)) is a perfect square.
%C A272798 Subsequence of A262406.
%C A272798 If n is a Carmichael number, then phi(n) = Product_{primes p dividing n} (p-1).
%C A272798 So the question is: What are the Carmichael numbers n such that Product_{primes p dividing n} (p-1) is a square?
%C A272798 The number of prime divisors of terms of this sequence are 3, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 5, 4, 4, 4, 4, 4, 4, 4, 4, 5, 4, 4, 4, 5, 5, 5, 4, 4, 4, 4, 4, ...
%C A272798 1299963601 = 601*1201*1801 is the second term that has three prime divisors and it is a member of this sequence since 600*1200*1800 = 2^10*3^4*5^6 is a square.
%C A272798 This sequence is infinite. See links section for more details. - _Altug Alkan_, Jan 16 2017
%H A272798 Amiram Eldar, <a href="/A272798/b272798.txt">Table of n, a(n) for n = 1..10000</a>
%H A272798 William D. Banks, <a href="http://dx.doi.org/10.4153/CMB-2009-001-7">Carmichael Numbers with a Square Totient</a>, Canad. Math. Bull. 52 (2009), 3-8.
%H A272798 <a href="/index/Ca#Carmichael">Index entries for sequences related to Carmichael numbers</a>.
%e A272798 1729 is a term because A000010(1729) = 1729*(1-1/7)*(1-1/13)*(1-1/19) = 1296 = 36^2.
%o A272798 (PARI) isA002997(n) = {my(f); bittest(n, 0) && !for(i=1, #f=factor(n)~, (f[2, i]==1 && n%(f[1, i]-1)==1)||return) && #f>1}
%o A272798 lista(nn) = for(n=1, nn, if(isA002997(n) && issquare(eulerphi(n)), print1(n, ", ")));
%Y A272798 Cf. A000010, A000290, A002997, A039770, A262406, A290793.
%K A272798 nonn
%O A272798 1,1
%A A272798 _Altug Alkan_, May 06 2016
%E A272798 a(30) corrected by _Amiram Eldar_, Aug 11 2017