OFFSET
1,1
COMMENTS
The integers with only one prime factor and whose cototient is a square are in A246551.
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.
LINKS
Harvey P. Dale, Table of n, a(n) for n = 1..1000
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.
EXAMPLE
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.
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 *)
PROG
(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
CROSSREFS
KEYWORD
nonn
AUTHOR
Bernard Schott, Feb 09 2019
STATUS
approved