cp's OEIS Frontend

A386838(k) is the minimal area of the graph formed under the requirement that the straight line drawn from (0,0) to (x,y) (where x^2 + y^2 = k = A001481(n)) passes through an enclosed space on the square lattice and its edges are either vertical or horizontal. If A001481(n) = x^2 + y^2 for multiple x and y, then x and y are chosen such that A386838(A001481(n)) is minimal. a(n) is the number of graphs with area n, and equivalently the number of numbers of the form x^2 + y^2 = A001481(n) such that n = x + y - gcd(x,y) for such minimal x and y.

The offset is 1 since 0 occurs infinitely many times in A386838 (e.g., A386838(k) = 0 when A001481(k) is square).

The range in which n can occur in A386838 is bounded above by 2*n^2.

Does every integer n > 0 appear in this sequence?

a(n) is the minimal area of the graph formed under the requirement that the straight line (0,0) to (x,y) passes through an enclosed space on the square lattice, with the graph drawn using only vertical and horizontal edges.

Every nonnegative integer n appears in this sequence. Proof: Since 2*n^2 = n^2 + n^2 then by the first formula in the formula section n + n - gcd(n,n) = n. To prove that a(m) = n when A001481(m) = 2*n^2, we have to prove that x = n and y = n is the choice such that a(m) is minimal. Let r and s be two other numbers such that 2*n^2 = r^2 + s^2. Let r > n: consequently s < n, 1 <= gcd(r,s) <= s, and s - gcd(r,s) >= 0. If r + s - gcd(r,s) <= n, then s - gcd(r,s) < 0. But s - gcd(r,s) >= 0. Therefore r + s - gcd(r,s) >= r > n, and a(m) = n.

Let G be a group of order n, let N = {1, 2, ..., n}, and let f: G -> N be a bijection whereby f(G) = I is an index set of G. An automorphism phi of G is a permutation of N via f(phi(G)). It is tempting to ask the question 'how many permutations of N obey the group laws?'. However this question is not well-defined since it would require there being a natural single choice of bijection for every group of order n, which in general does not exist. Enumerating permutations of N which are automorphisms for every isomorphism class G will therefore depend on the choice of bijection for G. a(n) is the upper bound for all such enumerations of permutations of size n since a(n) is either: the maximum enumeration when the choice of bijections ensures that all permutations are distinct; or a(n) is the enumeration including all multiplicities when the choice of bijections leads to permutations which are not distinct.

A086369(n) is also the cardinality of the set containing the divisors d of n and those 0 < m < n satisfying m + d = n.

Equivalently, orders k of groups G where a G exists as a direct product of isomorphic simple groups.

A group G is characteristically simple if it contains no characteristic proper subgroups (a subgroup which is invariant under every automorphism of G). Since a finite group is characteristically simple if and only if it is a direct product of isomorphic simple groups, G is characteristically simple if and only if it is an elementary abelian group or a direct product of isomorphic nonabelian simple groups.

a(n) is the lesser term of a pair of consecutive nonzero terms in A383844.

Triplets of consecutive nonzero terms can also be found in A383844 and are represented here as pairs. Up to n = 83354 there are 8 such triplets, the least terms of each being 6, 26, 27, 145, 577, 3042, 8683, 8857.

Numbers k such that there are exactly two m such that Sum_{i=1..t} m mod prime(i) for prime(t) <= m < prime(t+1) is equal to k (see A024934).

A383844(s) <= 3 for s <= 82000, with A383844(s) = 3 only for s = 0, 1, 8, 37, 781.

Numbers k such that there is only one m such that Sum_{i=1..t} m mod 2^i for 2^t <= m < 2^(t+1) is equal to k (see A049802). Every m = 2^k+1 (see A383327).

All terms are odd since for every even r there are at least two numbers u, v where the congruences sum to r (for u = 2^r+1 and v = 2^(r/2+1)+2).