# Find the next number in these series..?

1, 1, 1, 1, 2, 2, 2, 2, 2, 5, 4, ?

2, 4, 6, 10, 15, 26, ?

71, 194, 38, 1701, 89, 76, ?

5303944, 6677714, 2070936216988528558, ?

1, 2, 3, 6, 9, 9, 16, 20, 18, 30, ?

You know, there is more than finding the numbers itself, why not find another different number in the process?

Some of the sequences are difficult to determine independently so let us refer to our good friend

1, 1, 1, 1, 2, 2, 2, 2, 2, 5, 4, ?

This appears to be A115101 which is the number of distinct prime factors of $$L(n + F(n))$$ where $$F(n)$$ is the Fibonacci number and $$L(n)$$ is the Lucas number and $$n \geq 1$$. The next number is 7

2, 4, 6, 10, 15, 26, ?

This appears to be A113117 which is defined by $$a(1) = 2$$ and for $$n>1, a(n)$$ is the smallest integer $$> a(n-1)$$ such that all primes $$\leq a(n-1)$$ divide at least one integer $$k$$ for $$a(n-1) < k \leq a(n)$$. The next number is 46.

71, 194, 38, 1701, 89, 76, ?

This is A101110, the First Beale cipher. The next number is 11.

5303944, 6677714, 2070936216988528558, ?

This appears to be A116105 which are numbers $$k$$ such that $$k$$ concatenated with $$k-8$$ gives the product of two numbers which differ by $$5$$. The next number is 2969428172738875624

1, 2, 3, 6, 9, 9, 16, 20, 18, 30, ?

This is A097108 which is defined by "if a geodesic dome is made by dividing each triangle of an icosahedron into $$n^2$$ identical equilateral triangles and the vertices of those newly created triangles are pushed out from the center to lie on the surface of the sphere in which the icosahedron is inscribed, then this sequence gives the number of different strut lengths that are required to build the dome." The next number is 36

Hidden word

If we look at the sequence numbers on OEIS with the A removed, we have 115101, 113117, 101110, 116105, 097108. Notice how each of these numbers is a concatenation of two three-digit numbers close to 100.

If we split each number in half to make a new sequence 115, 101, 113, 117, 101, 110, 116, 105, 097, 108 and convert from decimal to ASCII, it spells sequential.

• no, no, you're right.. the final answer is a word. the said "another number" is the sequence numbers, and you dont need to find the numbers itself, i just see a lot of questions with this type of sequence
– poco
May 12, 2023 at 23:29
• @poco Ah, okay, thanks for clarifying. May 12, 2023 at 23:30