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

1729, 63973, 75361, 172081, 278545, 340561, 658801, 997633, 1773289, 3224065, 5310721, 8719309, 8719921, 12945745, 13187665, 15888313, 17586361, 27402481, 29020321, 39353665, 40430401, 49333201, 67371265, 84417985, 120981601, 128697361, 129255841, 130032865, 151530401, 151813201, 158864833

Offset: 1

Author

Altug Alkan, May 06 2016

Keywords

• nonn

Comments

Subsequence of A262406.

If n is a Carmichael number, then phi(n) = Product_{primes p dividing n} (p-1).

So the question is: What are the Carmichael numbers n such that Product_{primes p dividing n} (p-1) is a square?

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, ...

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.

This sequence is infinite. See links section for more details. - Altug Alkan, Jan 16 2017

Examples

			1729 is a term because A000010(1729) = 1729*(1-1/7)*(1-1/13)*(1-1/19) = 1296 = 36^2.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000
• William D. Banks, Carmichael Numbers with a Square Totient, Canad. Math. Bull. 52 (2009), 3-8.
• Index entries for sequences related to Carmichael numbers.

Crossrefs

Cf. A000010, A000290, A002997, A039770, A262406, A290793.

Programs

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}
lista(nn) = for(n=1, nn, if(isA002997(n) && issquare(eulerphi(n)), print1(n, ", ")));

Extensions

a(30) corrected by Amiram Eldar, Aug 11 2017