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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 (list; graph; refs; listen; history; text; internal format)
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
Intersection of A073539 and A151999. - Amiram Eldar, Jun 02 2020
Cf. A007947, A027748, A055742, A173557, A256248, subsequence of A124240.
Sequence in context: A312783 A312784 A070738 * A141718 A195382 A211413
KEYWORD
nonn
AUTHOR
Labos Elemer, Jul 11 2000
EXTENSIONS
Corrected and extended by James A. Sellers, Jul 11 2000
STATUS
approved

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Last modified March 2 06:32 EST 2024. Contains 370460 sequences. (Running on oeis4.)