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A265640 Prime factorization palindromes (see comments for definition). 5
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, 41, 43, 44, 45, 47, 48, 49, 50, 52, 53, 59, 61, 63, 64, 67, 68, 71, 72, 73, 75, 76, 79, 80, 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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

a(66) is the first term at which this sequence differs from A119848.

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.

Numbers which are both palindromes (A002113) and PFP are 1,2,3,4,5,7,9,11,44,99,101,... (see A265641).

If n is in the sequence, so is n^k for all k >= 0. - Altug Alkan, Dec 11 2015

The sequence contains all perfect numbers except 6 (cf. A000396). - Don Reble, Dec 12 2015

Equivalently, numbers that have at most one prime factor with odd multiplicity. - Robert Israel, Feb 03 2016

LINKS

Robert Israel, Table of n, a(n) for n = 1..10000

FORMULA

lim A(x)/pi(x) = zeta(2) where A(x) is the number of a(n) <= x and pi is A000720.

EXAMPLE

44 is a member, since 44=2*11*2.

180 is a member, since 180=2*3*5*3*2.

MAPLE

N:= 1000: # to get all terms <= N

P:= [1, op(select(isprime, [2, seq(i, i=3..N, 2)]))]:

sort([seq(seq(p*x^2, x=1..floor(sqrt(N/p))), p=P)]); # Robert Israel, Feb 03 2016

MATHEMATICA

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-Fran├žois Alcover, Apr 09 2019, after Robert Israel *)

PROG

(PARI) for(n=1, 200, if( ispseudoprime(core(n)) || issquare(n), print1(n, ", "))) \\ Altug Alkan, Dec 11 2015

CROSSREFS

Cf. A000396, A000720, A002113, A229125, A265641, complement of A229153.

Cf. A013661 (zeta(2)).

Sequence in context: A325370 A130091 A119848 * A268375 A048683 A231876

Adjacent sequences:  A265637 A265638 A265639 * A265641 A265642 A265643

KEYWORD

nonn

AUTHOR

Vladimir Shevelev, Dec 11 2015

STATUS

approved

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Last modified October 20 15:05 EDT 2019. Contains 328267 sequences. (Running on oeis4.)