A334110 The squares of squarefree numbers (A062503), ordered lexicographically according to their prime factors. a(n) = Product_{k in I} prime(k+1)^2, where I are the indices of nonzero binary digits in n = Sum_{k in I} 2^k.

1, 4, 9, 36, 25, 100, 225, 900, 49, 196, 441, 1764, 1225, 4900, 11025, 44100, 121, 484, 1089, 4356, 3025, 12100, 27225, 108900, 5929, 23716, 53361, 213444, 148225, 592900, 1334025, 5336100, 169, 676, 1521, 6084, 4225, 16900, 38025, 152100, 8281, 33124, 74529, 298116, 207025, 828100, 1863225, 7452900, 20449, 81796, 184041

Offset: 0

Author

Antti Karttunen and Peter Munn, May 01 2020

Keywords

• nonn

Comments

For the lexicographic ordering, the prime factors must be written in nonincreasing order. If we write the factors in nondecreasing order, we get a lexicographically ordered set with an order type that is greater than a natural number index - the resulting sequence does not include all qualifying numbers. (Note also that the symbols used for the lexicographic order are the prime numbers, not their digits.)

a(n) is the n-th power of 4 in the monoid defined in A331590.

Conjecture: a(n) is the position of the first occurrence of n in A334109.

Examples

			The initial terms are shown below, equated with the product of their prime factors to exhibit the lexicographic ordering. The list starts with 1, since 1 is factored as the empty product and the empty list is first in lexicographic order.
    1 = .
    4 = 2*2.
    9 = 3*3.
   36 = 3*3*2*2.
   25 = 5*5.
  100 = 5*5*2*2.
  225 = 5*5*3*3.
  900 = 5*5*3*3*2*2.
   49 = 7*7.
  196 = 7*7*2*2.
  441 = 7*7*3*3.
		

Crossrefs

Cf. A000079, A019565 (square roots), A048675, A097248, A225546, A267116, A332382, A334109 (a left inverse).

Column 2 of A329332. Permutation of A062503.

After 1, the right children of the leftmost edge of A334860, or respectively, the left children of the rightmost edge of A334866.

Subsequences: A001248, A061742, A166329.

Subsequence of A052330.

A003961, A003987, A059897, A331590 are used to express relationship between terms of this sequence.

Programs

Mathematica

Array[If[# == 0, 1, Times @@ Flatten@ Map[Function[{p, e}, Map[Prime[Log2@ # + 1]^(2^(PrimePi@ p - 1)) &, DeleteCases[NumberExpand[e, 2], 0]]] @@ # &, FactorInteger[3^#]]] &, 51, 0] (* Michael De Vlieger, May 26 2020 *)

PARI

A334110(n) = { my(p=2,m=1); while(n, if(n%2, m *= p^2); n >>= 1; p = nextprime(1+p)); (m); };

Formula

a(n) = A019565(n)^2.

For n >= 1, a(A000079(n-1)) = A001248(n).

For all n >= 0, A334109(a(n)) = n.

a(n+k) = A331590(a(n), a(k)).

a(n XOR k) = A059897(a(n), a(k)), where XOR denotes bitwise exclusive-or, A003987.

a(n) = A225546(3^n).

a(2n) = A003961(a(n)).

a(2n+1) = 4 * a(2n).

a(2^k-1) = A061742(k).

A267116(a(n)) = 2.

A048675(a(n)) = 2n.

A097248(a(n)) = A332382(n) = A019565(2n).