A306670 Numbers k with exactly three distinct prime factors and such that cototient(k) is a square.

345, 465, 468, 1332, 1545, 1833, 1872, 2628, 2737, 2769, 3105, 3145, 3585, 3657, 3945, 4081, 4100, 4185, 4212, 4345, 5328, 6465, 6516, 6785, 6945, 7105, 7488, 8428, 8569, 8625, 8961, 10257, 10512, 10785, 10833, 10945, 11625, 11988, 12132, 12865

Offset: 1

Author

Bernard Schott, Mar 04 2019

Keywords

• nonn

Comments

The integers with only one prime factor and whose cototient is a square are in A246551. The integers with two prime factors and whose cototient is a square are in A323916, and the subsequences A323917 and A323918.

There are exactly three different families of integers which realize a partition of this sequence. See the file "Subfamilies and subsequences" (& III) in A063752 for more details, proofs with data, comments, formulas and examples.

Examples

			1st family: 2769 = 3 * 13 * 71 and cototient(2769) = 33^2.
2nd family: 14841 = 3^2 * 17 * 97 and cototient(14841) = 75^2.
3rd family: 1872 = 2^4 * 3^2 * 13 and cototient(1872) = 36^2.
		

Crossrefs

Subsequence of A063752.

Cf. A246551 (only one prime factor), A323916, A323917, A323918 (two prime factors), A000396 (even perfect numbers).

Formula

1st family: The primitive terms are p*q*r with p,q,r primes and cototient(p*q*r) = p*q*r-(p-1)*(q-1)*(r-1) = M^2. These primitives generate the entire family formed by the numbers k = p^(2s+1) * q^(2t+1) * r^(2u+1) with s,t,u >=0, and cototient(k) = (p^s * q^t * r^u * M)^2.

2nd family: The primitive terms are p^2 *q * r with p,q,r primes and cototient(p^2 * q * r) = p * (p*q*r-(p-1)*(q-1)*(r-1)) = M^2. These primitives generate the entire family formed by the numbers k = p^(2s) * q^(2t+1) * r^(2u+1) with s>=1, t,u >=0, and cototient(k) = (p^(s-1) * q^t * r^u * M)^2.

3rd family: The primitive terms are p^2 * q^2 * r with p,q,r primes and cototient(p^2 * q^2 * r) = p * q * (p*q*r-(p-1)*(q-1)*(r-1)) = M^2. These primitives generate the entire family formed by the numbers k = p^(2s) * q^(2t) * r^(2u+1) with s,t>=1, u >=0, and cototient(k) = (p^(s-1) * q^(t-1) * r^u * M)^2.