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A055744
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Numbers k such that k and phi(k) have the same prime factors.
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22
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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
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1,2
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COMMENTS
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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.
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
Pollack and Pomerance call these numbers "phi-perfect numbers". - Amiram Eldar, Jun 02 2020
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LINKS
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Paul Pollack and Carl Pomerance, Prime-Perfect Numbers, INTEGERS, Electronic J. of Combinatorial Number Theory, Vol. 12a, Paper A14, 2012.
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EXAMPLE
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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.
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MAPLE
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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(n-1)+1 do
if isA055744(a) then
return a;
end if;
end do:
end if;
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MATHEMATICA
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Select[Range@ 1800,
First /@ FactorInteger@ # == First /@ FactorInteger@ EulerPhi@ # &] (* Michael De Vlieger, Mar 21 2015 *)
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PROG
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(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
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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