A366170 Lexicographically earliest sequence of distinct positive integers such that for n>1, Sum_{i=1..n, a(i)<=n} a(a(i)) is prime.

1, 2, 4, 8, 6, 12, 5, 7, 10, 18, 13, 11, 14, 16, 17, 24, 20, 22, 21, 15, 19, 26, 3, 23, 9, 30, 28, 32, 31, 29, 40, 42, 34, 36, 37, 44, 35, 27, 41, 50, 46, 39, 33, 56, 47, 52, 68, 49, 43, 54, 53, 51, 60, 58, 57, 45, 66, 55, 61, 74, 64, 63, 84, 72, 67, 78, 65, 59, 70, 90, 73, 80

Offset

1,2

Comments

At a new term k, a(n) = k adds a(k) to the current prime sum if k <= n. If n is a term in the sequence among a(1..n-1), then a(n) = k is added. If neither of these conditions is met, the current prime sum remains the same.

If k is even and a(k) odd, then k cannot appear as a(n) = k at any n >= k (otherwise, the intended prime sum will be even, and thus not prime). This means that some even numbers will miss their chance and never appear. 38 is the smallest missing number.

Can it be proved that every odd number appears?

Links

Neal Gersh Tolunsky, Table of n, a(n) for n = 1..10000

Neal Gersh Tolunsky, Graph of the first differences for n = 1..10001

Example

At [1,2], the terms at indices i=1 and i=2, namely 1 and 2, sum to 3, a prime.
At [1,2,4], i=4 is not the index of a term in the sequence yet, so the sum remains the same.
At [1,2,4,8], the sum of the terms at i=1,2,4 is a(1)=1 + a(2)=2 + a(4)=8, which is 11, a prime number.

Crossrefs

Cf. A054408, A180405, A359535.

Sequence in context: A249681 A256561 A125597 * A151732 A109554 A339821

Adjacent sequences: A366167 A366168 A366169 * A366171 A366172 A366173

Keyword

nonn

Author

Neal Gersh Tolunsky, Oct 02 2023

Status

approved