A323916 Numbers k with exactly two distinct prime divisors and such that cototient(k) is a square.

6, 21, 24, 28, 54, 68, 69, 96, 112, 124, 133, 141, 189, 216, 237, 272, 284, 301, 384, 388, 448, 481, 486, 496, 501, 508, 589, 621, 657, 669, 781, 796, 864, 964, 1025, 1029, 1077, 1088, 1136, 1141, 1269, 1317, 1348, 1357, 1372, 1417, 1536, 1537, 1552, 1701, 1792, 1796

Offset: 1

Author

Bernard Schott, Feb 09 2019

Keywords

• nonn

Comments

The integers with only one prime factor and whose cototient is a square are in A246551.

This sequence is the intersection of A007774 and A063752.

There are exactly two different families of integers which realize a partition of this sequence (A323917 and A323918); there is also another family with the even perfect numbers of A000396 which is a subsequence of this sequence.

See the file "Subfamilies of terms" (& II) in A063752 for more details, proofs with data, comments, formulas and examples.

Examples

			1st family: 189 = 3^3 * 7 and cototient(189) = 9^2;
2nd family: 272 = 2^4 * 17 and cototient(272) = 12^2;
3rd family: 8128 = 2^6 * 127 and cototient(8128) = 64^2.
		

Links

• Harvey P. Dale, Table of n, a(n) for n = 1..1000

Crossrefs

Subsequence of A063752.

Cf. A051953.

Cf. A000396, A323917, A323918

Programs

Mathematica

Select[Range[1800], 2 == Length@ FactorInteger@ # == 2 && IntegerQ@ Sqrt[# - EulerPhi@ #] &] (* Giovanni Resta, Feb 27 2019 *)
Select[Range[2000],PrimeNu[#]==2&&IntegerQ[Sqrt[#-EulerPhi[#]]]&] (* Harvey P. Dale, Jan 06 2022 *)

PARI

isok(n) = (omega(n)==2) && issquare(n - eulerphi(n)); \\ Michel Marcus, Feb 10 2019

Sage

[n for n in (1..2500) if len([1 for d in divisors(n) if is_prime(d)])==2 and is_square(n - euler_phi(n))] # G. C. Greubel, Mar 01 2019

Formula

1st family (A323917): if k = p^(2s+1) * q^(2t+1) with s,t >=0, p, q primes and p + q - 1 = M^2, then cototient(k) = (p^r * q^s * M)^2. The primitive terms are p*q with cototient(p*q) = p+q-1 = M^2

2nd family (A323918): if k = p^(2s) * q^(2t+1) with s>=1, t>=0, p, q primes, p < q and such that p*(p+q-1)= M^2, then cototient(k) = (p^(s-1) * q^t * M)^2. The primitive terms are p^2 *q with cototient(p^2 * q) = p * (p+q-1) = M^2

3rd family (A000396): the even perfect Numbers, if 2^p - 1 is a Mersenne prime, then cototient(2^(p-1) * (2^p - 1)) = (2^(p-1))^2.