cp's OEIS Frontend

Theorem: Indices of even terms give A028982. Indices of odd terms give A028983.

If A001227(n) is odd then a(n) is even.

If A001227(n) is even then a(n) is odd.

The above sentences arise that the diagram is always symmetric for any value of n hence the number of edges is always an even number. Also from Euler's formula.

For another version see A340833 from which first differs at a(6).

For the definition of subparts see A279387. For more information about the subparts see also A237271, A280850, A280851, A296508, A335616.

Note that in this version of the diagram of the symmetric representation of sigma(n) all regions are called "subparts". The number of subparts equals A001227(n).

The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.

We define the alternating product of a sequence (y_1,...,y_k) to be Product_i y_i^((-1)^(i-1)).

Also numbers whose multiset of prime indices has non-integer reverse-alternating product.

For any hypothetical quasiperfect number q (for which sigma(q) = 2*q+1, see A336701), a(q) would be equal to 2*q XOR 2*q+2 = 2*(q XOR q+1) = 2*A038712(1+q) = A100892(1+q).

See also A000079 and A235796 concerning the "almost perfect" or "least deficient" numbers that give positions of 0's here.

All odd primes are in the sequence.

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)

Numbers k such that A187793(k) = A187794(k) + A187795(k).

All the terms are nondeficient numbers (A023196).

All the perfect numbers (A000396) are terms.

This sequence is infinite: if k = 2^(p-1)*(2^p-1) is an even perfect number and q > 2^p-1 is a prime, then k*q is a term.

Since the total sum of divisors of any term is even, none of the terms are squares or twice squares.

Are there odd terms in this sequence? There are none below 10^10.

The numbers of terms not exceeding 10^k, for k = 1, 2, ..., are 1, 6, 63, 605, 6164, 61291, 614045, 6139193, 61382607, 613861703, ... . Apparently, the asymptotic density of this sequence exists and equals 0.06138... .

Practical numbers without any squares or twice squares.

The subsequence of odd terms in A274397.

If n is in the sequence and gcd(n,m)=1 for some odd m, then n*m is also in the sequence. One might call "primitive" those terms which are not of this form, i.e., not a "coprime" multiple of an earlier term. The list of these primitive terms is (19, 27, 29, 59, 79, 89, 109, 139, 149, 179, 199, 229, 239, 269, 343, 349, 359, 379, 389, 409, 419, 439, 449, 479, 499, ...). The primitive terms are the primes and powers of primes within the sequence. If a prime power p^k (k >= 1) is in the sequence, then p^(m(k+1)-1) is in the sequence for any m >= 1, since 1+p+...+p^(m(k+1)-1) = (1+p+...+p^k)(1+p^(k+1)+...+p^((m-1)*(k+1))). For example, with the prime p=19 we also have all odd powers 19^3, 19^5, ..., and with 27 = 3^3, we also have 27^5, 27^9, ... in the sequence.

On the other hand, for any prime p <> 5 there is an exponent k in {1, 3, 4} such that p^k is in this sequence (and therewith all higher powers of the form given above).

One may notice that there are many pairs of the form (30k-3, 30k-1), e.g., 27,29; 57,59; 87,89; 177,179; 237,239; 295,299; ... Indeed, it is likely that 30k-1 is prime and in this case, if 10k-1 is also prime, then sigma(30k-3) = 40k is divisible by 5 and sigma(30k-1) = 30k is also divisible by 5.