OFFSET
1,2
COMMENTS
Contains products of suitable powers of 2 and Fermat primes. For x = 2^u*3^w, phi(x) = 2^u*3^(w-1) with suitable exponents. Analogous constructions are possible with {2,3,7} prime divisors, etc.
From Ivan Neretin, Mar 19 2015: (Start)
Also, numbers k that meet the following criteria for every prime p dividing k:
1. All prime divisors of p-1 must also divide k;
2. If k has no prime divisors of the form m*p+1, and k is divisible by p, then k must be divisible by p^2.
Also, numbers k for which {k, phi(k), phi(phi(k))} is a geometric progression.
(End)
All terms > 1 are even. An even number k is in the sequence iff 2*k is in the sequence. - Robert Israel, Mar 19 2015
For n > 1, the largest prime factor of a(n) has multiplicity >= 2. For all prime factors more than half of the largest prime factor of a(n), the multiplicity differs from 1.
If k = p1^a1 * p2^a2 * ... * pm^am is in the sequence, then so is p1^b1 * p2^b2 * ... * pm^bm for 1 <= i <= m and prime pi and bi >= ai.
If m * p^2 is not in the sequence, for a prime p and some m > 0, then neither is m * p^3. - David A. Corneth, Mar 22 2015
A027748(a(n),j) = A027748(A000010(a(n)),j) for j=1..A001221(a(n)); also numbers k such that k and phi(k) have the same squarefree kernel: A007947(a(n)) = A007947(A000010(a(n))). - Reinhard Zumkeller, Jun 01 2015
Pollack and Pomerance call these numbers "phi-perfect numbers". - Amiram Eldar, Jun 02 2020
LINKS
David A. Corneth, Table of n, a(n) for n = 1..117561 (terms <= 10^11; first 10000 terms from T. D. Noe)
Paul Pollack and Carl Pomerance, Prime-Perfect Numbers, INTEGERS, Electronic J. of Combinatorial Number Theory, Vol. 12a, Paper A14, 2012.
EXAMPLE
k = 578 = 2*17*17, phi(578) = 272 = 2*2*2*2*17 with 2 and 17 prime factors, so 578 is a term.
k = 588 = 2*2*3*7*7, phi(588) = 168 = 2*2*2*3*7, so 588 is a term.
k = 264196 = 2*2*257*257, phi(264196) = 512*257 = 131584, so 264196 is a term.
MAPLE
select(numtheory:-factorset = numtheory:-factorset @ numtheory:-phi,
[1, 2*i $ i=1..2000]); # Robert Israel, Mar 19 2015
isA055744 := proc(n)
nfs := numtheory[factorset](n) ;
phinfs := numtheory[factorset](numtheory[phi](n)) ;
if nfs = phinfs then
true;
else
false;
end if;
end proc:
A055744 := proc(n)
if n = 1 then
1;
else
for a from procname(n-1)+1 do
if isA055744(a) then
return a;
end if;
end do:
end if;
end proc: # R. J. Mathar, Sep 23 2016
MATHEMATICA
Select[Range@ 1800,
First /@ FactorInteger@ # == First /@ FactorInteger@ EulerPhi@ # &] (* Michael De Vlieger, Mar 21 2015 *)
PROG
(PARI) is(n)=factor(n)[, 1]==factor(eulerphi(n))[, 1] \\ Charles R Greathouse IV, Oct 31 2011
(PARI) is(n)=my(f=factor(n)); f[, 1]==factor(eulerphi(f))[, 1] \\ Charles R Greathouse IV, May 26 2015
(Haskell)
a055744 n = a055744_list !! (n-1)
a055744_list = 1 : filter f [2..] where
f x = all ((== 0) . mod x) (concatMap (a027748_row . subtract 1) ps) &&
all ((== 0) . mod (a173557 x))
(map fst $ filter ((== 1) . snd) $ zip ps $ a124010_row x)
where ps = a027748_row x
-- Reinhard Zumkeller, Jun 01 2015
CROSSREFS
KEYWORD
nonn
AUTHOR
Labos Elemer, Jul 11 2000
EXTENSIONS
Corrected and extended by James A. Sellers, Jul 11 2000
STATUS
approved