cp's OEIS Frontend

Essentially the same as A062312: a(1)=0, a(n)=A062312(n-1) for n>=2. - R. J. Mathar, Sep 11 2012

All values of this sequence are nonsquarefree (A013929).

From Peter Munn, Nov 23 2020: (Start)

Numbers whose square part is greater than 4. [Proof follows from s and t having to be multiples of A019554(k), the smallest number whose square is divisible by k.]

Compare with A116451, numbers whose odd part is greater than 3. The self-inverse function A225546(.) maps the members of either one of these sets 1:1 onto the other set.

Compare with A028983, numbers whose squarefree part is greater than 2.

(End)

The asymptotic density of this sequence is 1 - 15/(2*Pi^2). - Amiram Eldar, Mar 08 2021

From Bernard Schott, Jan 09 2022: (Start)

Numbers of the form u*m^2, for u >= 1 and m >= 3 (union of first 2 comments).

A geometric application: in trapezoid ABCD, with AB // CD, the diagonals intersect at E. If the area of triangle ABE is u and the area of triangle CDE is v, with u>v, then the area of trapezoid ABCD is w = u + v + 2*sqrt(u*v); in this case, u, v, w are integer solutions iff (u,v,w) = (k*s^2,k*t^2,k*(s+t)^2), with s>t and k positives; hence, w is a term of this sequence (see IMTS link). (End)

Subsequence of A072413 and differs from it by not having the terms 216, 1000, 1080, 1512, ... .

Each term can be represented in a unique way as m * k^2, where m is an exponentially odd number (A268335) and k is a composite number that is coprime to m.

Numbers k such that A350388(k) is a square of a composite number (A062312 \ {1}).

The asymptotic density of this sequence is 1 - Product_{p prime} (1 - 1/(p*(p+1))) * (1 + Sum_{p prime} 1/(p^2+p-1)) = 0.032993560887093165933... .

From Jon E. Schoenfield, Feb 02 2018: (Start)

Let S(j) be the partial sum through the j-th term of the alternating series, i.e., S(j) = -Sum_{m=1..j} (-1)^m/A002808(m)^2. The sequence of real values S(2*i-1) for i >= 1, i.e., of partial sums 1/16, 1/16 - 1/36 + 1/64, 1/16 - 1/36 + 1/64 - 1/81 + 1/100, ... (each of which ends with a positive term) will approach the limit from above, while the sequence of real values S(2*i) for i >= 1, i.e., of partial sums 1/16 - 1/36, 1/16 - 1/36 + 1/64 - 1/81, 1/16 - 1/36 + 1/64 - 1/81 + 1/100 - 1/144, ... (each of which ends with a negative term) will approach the limit from below. Let S'(j) = (S(j-1) + S(j))/2; equivalently, S'(j) = -(Sum_{m=1..j-1} (-1)^m/A002808(m)^2 + (1/2)*(-1)^j/A002808(j)^2), so S'(j) can be viewed as an adjusted version of S(j), adjusted by using only half of the final term of S(j). At large values of j, successive values of S'(j) will fluctuate very little compared to the differences between successive values of S(j), because the averaging of successive values of S(j), which are above the limit at each odd value of j and below the limit by very nearly the same amount at each even value of j, causes the values of S'(j) to trace a path midway between that traced by the S(j) values for odd j and those for even j.

Moreover, similar to the situation at A275712, it can be verified that the values of S'(j) themselves fall into three sharply distinct real-valued subsequences: one that converges toward the limit from above and consists of those values where both j and the j-th composite number (i.e., the square root of the reciprocal of the last term in S(j)) are even; one that converges toward the limit from below and consists of those values where j is odd and the j-th composite number is even; and one that stays very near the middle, converging even more rapidly toward the limit, and consisting of all those values where the j-th composite number is odd (regardless of the parity of j). The values in this last subsequence converge very rapidly; see the table in the Example section, which lists values of S'(c_k) where c_k is the smallest odd composite number > 2^k. (End)

The odd primes squared plus 1 and the nonprimes squared minus 1 are the numerators or denominators of an infinite product converging to 1 whose denominators or numerators, conversely, are the squares of said numbers, that is, (p^2+1/p^2)*(q^2-1)/q^2)..., where p is an odd prime and q is a nonprime.

Union of A066872 and (A062312 - 1) with 0 and 5 removed. - Robert Israel, Aug 26 2014

This sequence is the second subsequence of A326707: squares of composites which have no Brazilian representation with three digits or more.

As tau(m) = 2 * beta(m) + 3, the number of divisors of these squares of composites m is odd with tau(m) >= 5.

The corresponding composites are: 4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 21, 22, 24, 25, 26, 27, 28, 30, 32, 33, 34, 35, 36, 38, 42, ...