

A055744


Numbers k such that k and phi(k) have the same prime factors.


22



1, 4, 8, 16, 18, 32, 36, 50, 54, 64, 72, 100, 108, 128, 144, 162, 200, 216, 250, 256, 288, 294, 324, 400, 432, 450, 486, 500, 512, 576, 578, 588, 648, 800, 864, 882, 900, 972, 1000, 1014, 1024, 1152, 1156, 1176, 1210, 1250, 1296, 1350, 1458, 1600, 1728, 1764
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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^(w1) with suitable exponents. Analogous constructions are possible with {2,3,7} prime divisors, etc.
Also, numbers k that meet the following criteria for every prime p dividing k:
1. All prime divisors of p1 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
Pollack and Pomerance call these numbers "phiperfect numbers".  Amiram Eldar, Jun 02 2020


LINKS

Paul Pollack and Carl Pomerance, PrimePerfect 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,
isA055744 := proc(n)
nfs := numtheory[factorset](n) ;
phinfs := numtheory[factorset](numtheory[phi](n)) ;
if nfs = phinfs then
true;
else
false;
end if;
end proc:
if n = 1 then
1;
else
for a from procname(n1)+1 do
if isA055744(a) then
return a;
end if;
end do:
end if;


MATHEMATICA

Select[Range@ 1800,
First /@ FactorInteger@ # == First /@ FactorInteger@ EulerPhi@ # &] (* Michael De Vlieger, Mar 21 2015 *)


PROG

(Haskell)
a055744 n = a055744_list !! (n1)
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


CROSSREFS



KEYWORD

nonn


AUTHOR



EXTENSIONS



STATUS

approved



