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%I #61 Jul 11 2020 23:43:26
%S 1,2,3,4,5,7,8,9,11,12,13,16,17,18,19,20,23,25,27,28,29,31,32,36,37,
%T 41,43,44,45,47,48,49,50,52,53,59,61,63,64,67,68,71,72,73,75,76,79,80,
%U 81,83,89,92,97,98,99,100,101,103,107,108,109,112,113,116,117,121,124,125,127,128,131,137,139,144
%N Prime factorization palindromes (see comments for definition).
%C a(66) is the first term at which this sequence differs from A119848.
%C A number N is called a prime factorization palindrome (PFP) if all its prime factors, taking into account their multiplicities, can be arranged in a row with central symmetry (see example). It is easy to see that every PFP-number is either a square or a product of a square and a prime. In particular, the sequence contains all primes.
%C Numbers which are both palindromes (A002113) and PFP are 1,2,3,4,5,7,9,11,44,99,101,... (see A265641).
%C If n is in the sequence, so is n^k for all k >= 0. - _Altug Alkan_, Dec 11 2015
%C The sequence contains all perfect numbers except 6 (cf. A000396). - _Don Reble_, Dec 12 2015
%C Equivalently, numbers that have at most one prime factor with odd multiplicity. - _Robert Israel_, Feb 03 2016
%C Numbers whose squarefree part is noncomposite. - _Peter Munn_, Jul 01 2020
%H Robert Israel, <a href="/A265640/b265640.txt">Table of n, a(n) for n = 1..10000</a>
%F lim A(x)/pi(x) = zeta(2) where A(x) is the number of a(n) <= x and pi is A000720.
%e 44 is a member, since 44=2*11*2.
%e 180 is a member, since 180=2*3*5*3*2.
%p N:= 1000: # to get all terms <= N
%p P:= [1,op(select(isprime, [2,seq(i,i=3..N,2)]))]:
%p sort([seq(seq(p*x^2,x=1..floor(sqrt(N/p))),p=P)]); # _Robert Israel_, Feb 03 2016
%t M = 200; P = Join[{1}, Select[Join[{2}, Range[3, M, 2]], PrimeQ]]; Sort[ Flatten[Table[Table[p x^2, {x, 1, Floor[Sqrt[M/p]]}], {p, P}]]] (* _Jean-François Alcover_, Apr 09 2019, after _Robert Israel_ *)
%o (PARI) for(n=1, 200, if( ispseudoprime(core(n)) || issquare(n), print1(n, ", "))) \\ _Altug Alkan_, Dec 11 2015
%Y Cf. A000396, A000720, A002113, A265641, complement of A229153.
%Y Disjoint union of A229125 and (A000290\{0}).
%Y Cf. A013661 (zeta(2)).
%K nonn
%O 1,2
%A _Vladimir Shevelev_, Dec 11 2015
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