cp's OEIS Frontend

For n>1 these are odd composite numbers: all terms a(n) are divisible by number h(n) = GCD(n^5+n+1,(n+1)^5+n) = GCD(a(n), a(n+1)-2) = (n*(n+1)+1)*GCD(n*(n+1)-1, 5) where 1 < h(n) < a(n) for all n>1. Sequence of corresponding numbers h(n) for n>1: 35, 13, 21, 31, 43, 285, ... For example, a(7) = 16815 is divisible by number h(7) = (7*(7+1)+1)*GCD(7*(7+1)-1, 5) = 57*GCD(55, 5) = 57*5 = 285.

We name a set of k sequences IOPR_k(n) = {a_1(n) = a(n), a_2(n) = a(n) + 2, ..., a_k(n) = a(n) + 2*(k - 1)} as infinite nonprime k-lane road if a arithmetic function a(n) defined by arithmetic operations produces for all n > h (h = a small integer >= 0) odd terms such that all values a(n), a(n) + 2, ..., a(n) + 2*(k - 1) are composites. We say sequences a_1(n) = a(n), a_2(n) = a(n) + 2, ..., a_k(n) = a(n) + 2*(k - 1) are k-th lanes of set IOPR_k(n).

For example, sequence A016945(n) = 6*n + 3 = IOPR_1(n) for k=1.

This sequence a(n) is 2nd lane of set of sequences IOPR_2(n) = {a_1(n) = A271208(n) = a(n) - 2 = n^5 + n - 1, a_2(n) = a(n) = n^5 + n + 1}.

If p = prime > 2 of the form 3m - 1 from A003627 then sets of 2 sequences {n^p + n - 1, n^p + n + 1} = IOPR_2(n) for all p.

Also sets of 2 sequences {n^k + n - 1, n^k + n + 1} = IOPR_2(n) for all k>2 from A016789.

In general, if k>2 is number of the form 3m - 1 from A016789 then sequences a(n) = n^k + n - 1 and b(n) = a(n) + 2 = n^k + n + 1 produces for all n > 1 odd composite terms. The terms of sequence a(n) = n^k + n - 1 are divisible for all n > 1 by number h(n) = GCD(n^k+n-1,(n-1)^k+n) = GCD(a(n), a(n-1)+2) = (n*(n-1)+1)*GCD(n*(n-1)-1, k) where 1 < h(n) < a(n) for all n>1. The terms of sequence b(n) = a(n) + 2 = n^k + n + 1 are divisible for all n > 1 by number h(n) = GCD(n^k+n+1,(n+1)^k+n) = GCD(a(n), a(n+1)-2) = (n*(n+1)+1)*GCD(n*(n+1)-1, k) where 1 < h(n) < a(n) for all n>1.

Are there any sets of sequences IOPR_k(n) for k>2? For example, like set of sequences {A161945(n), A161945(n) + 2, A161945(n) + 4} is not an infinite nonprime 3-lane road because sequence A161945 is not defined by arithmetic operations.

a(n) = n^7 + n, A005843 for k=1, A002378 for k=2, A034262 for k=3, A091940 for k=4, A131471 for k=5, A131472 for k=6.

The idea of this sequence comes from the 6th problem of the 21st British Mathematical Olympiad in 1985 where it is asked to show that this equation has infinitely many solutions (see link Olympiads and reference Gardiner).

Indeed, this Diophantine equation x^2 + y^2 = z^5 + z with gcd(x, y, z) = 1 has solutions iff z is in A008784.

When z is in A008784, there exist (u, v), gcd(u, v) = 1 such that z = u^2 + v^2; then, (u*z^2-v)^2 + (u+v*z^2)^2 = z^5 + z. Hence, with x = min(u*z^2-v, u+v*z^2) and y = max(u*z^2-v, u+v*z^2), the equation x^2 + y^2 = z^5 + z is satisfied. So this equation has infinitely many solutions since it has at least one solution for each term of A008784.

For instance, for z = 10 we have:

with (u,v) = (1,3), then x = 1*10^2-3 = 97 and y = 1+3*10^2 = 301,

with (u,v) = (3,1), then x = 3+1*10^2 = 103 and y = 3*10^2-1 = 299,

so finally 97^2 + 301^2 = 103^2 + 299^2 = 10^5 + 10.

Note that some z, among them 10, have other solutions not of this form.

The name expresses the 'row view' of the array. The 'column view' regards the array as the collection of the inverse Möbius transforms of the power sequences k^n = 0^n, 1^n, 2^n, .... (n >= 0). Viewed this way, the array is a generalization of the number of divisors sequence tau (A000005), to which it reduces in the case k = 1.

The array has offset (0, 0). It uses the usual definition of 'k divides n' as described in Apostol, rather than the shortened version, which restricts to values k > 0 as some programs do (but not SageMath). Such a restriction makes sense in the context of rational numbers but not in the case of natural numbers.