cp's OEIS Frontend

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.

%I A334110 #40 Jun 11 2020 23:43:11
%S A334110 1,4,9,36,25,100,225,900,49,196,441,1764,1225,4900,11025,44100,121,
%T A334110 484,1089,4356,3025,12100,27225,108900,5929,23716,53361,213444,148225,
%U A334110 592900,1334025,5336100,169,676,1521,6084,4225,16900,38025,152100,8281,33124,74529,298116,207025,828100,1863225,7452900,20449,81796,184041
%N 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.
%C A334110 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.)
%C A334110 a(n) is the n-th power of 4 in the monoid defined in A331590.
%C A334110 Conjecture: a(n) is the position of the first occurrence of n in A334109.
%F A334110 a(n) = A019565(n)^2.
%F A334110 For n >= 1, a(A000079(n-1)) = A001248(n).
%F A334110 For all n >= 0, A334109(a(n)) = n.
%F A334110 a(n+k) = A331590(a(n), a(k)).
%F A334110 a(n XOR k) = A059897(a(n), a(k)), where XOR denotes bitwise exclusive-or, A003987.
%F A334110 a(n) = A225546(3^n).
%F A334110 a(2n) = A003961(a(n)).
%F A334110 a(2n+1) = 4 * a(2n).
%F A334110 a(2^k-1) = A061742(k).
%F A334110 A267116(a(n)) = 2.
%F A334110 A048675(a(n)) = 2n.
%F A334110 A097248(a(n)) = A332382(n) = A019565(2n).
%e A334110 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.
%e A334110     1 = .
%e A334110     4 = 2*2.
%e A334110     9 = 3*3.
%e A334110    36 = 3*3*2*2.
%e A334110    25 = 5*5.
%e A334110   100 = 5*5*2*2.
%e A334110   225 = 5*5*3*3.
%e A334110   900 = 5*5*3*3*2*2.
%e A334110    49 = 7*7.
%e A334110   196 = 7*7*2*2.
%e A334110   441 = 7*7*3*3.
%t A334110 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 *)
%o A334110 (PARI) A334110(n) = { my(p=2,m=1); while(n, if(n%2, m *= p^2); n >>= 1; p = nextprime(1+p)); (m); };
%Y A334110 Cf. A000079, A019565 (square roots), A048675, A097248, A225546, A267116, A332382, A334109 (a left inverse).
%Y A334110 Column 2 of A329332. Permutation of A062503.
%Y A334110 After 1, the right children of the leftmost edge of A334860, or respectively, the left children of the rightmost edge of A334866.
%Y A334110 Subsequences: A001248, A061742, A166329.
%Y A334110 Subsequence of A052330.
%Y A334110 A003961, A003987, A059897, A331590 are used to express relationship between terms of this sequence.
%K A334110 nonn
%O A334110 0,2
%A A334110 _Antti Karttunen_ and _Peter Munn_, May 01 2020