Sequence Definition
Construct a sequence of positive integers a(n)
as follows:
a(0) = 4
- Each term
a(n)
, other than the first, is the smallest number that satisfies the following:
a)a(n)
is a composite number,
b)a(n) > a(n-1)
, and
c)a(n) + a(k) + 1
is a composite number for each0 <= k < n
.
So we start with a(0) = 4
. The next entry, a(1)
must be 9
. It can't be 5
or 7
since those aren't composite, and it can't be 6
or 8
because 6+4+1=11
is not composite and 8+4+1=13
is not composite. Finally, 9+4+1=14
, which is composite, so a(1) = 9
.
The next entry, a(2)
must be 10
, since it's the smallest number larger than 9
with 10+9+1=20
and 10+4+1=15
both composite.
For the next entry, 11
and 13
are both out because they're not composite. 12
is out because 12+4+1=17
which is not composite. 14
is out because 14+4+1=19
which is not composite. Thus, 15
is the next term of the sequence because 15
is composite and 15+4+1=20
, 15+9+1=25
, and 15+10+1=26
are all each composite, so a(3) = 15
.
Here are the first 30 terms in this sequence:
4, 9, 10, 15, 16, 22, 28, 34, 35, 39, 40, 46, 52, 58, 64, 70, 75, 76, 82, 88, 94, 100, 106, 112, 118, 119, 124, 125, 130, 136
This is OEIS A133764.
Challenge
Given an input integer n
, output the n
th term in this sequence.
Rules
- You can choose either 0- or 1-based indexing. Please state which in your submission.
- The input and output can be assumed to fit in your language's native integer type.
- The input and output can be given by any convenient method.
- Either a full program or a function are acceptable. If a function, you can return the output rather than printing it.
- Standard loopholes are forbidden.
- This is code-golf so all usual golfing rules apply, and the shortest code (in bytes) wins.