A305801 - OEIS

A305801

Lexicographically earliest infinite sequence such that a(i) = a(j) => f(i) = f(j), where f(n) = 0 if n is an odd prime, with f(n) = n for all other n.

1, 2, 3, 4, 3, 5, 3, 6, 7, 8, 3, 9, 3, 10, 11, 12, 3, 13, 3, 14, 15, 16, 3, 17, 18, 19, 20, 21, 3, 22, 3, 23, 24, 25, 26, 27, 3, 28, 29, 30, 3, 31, 3, 32, 33, 34, 3, 35, 36, 37, 38, 39, 3, 40, 41, 42, 43, 44, 3, 45, 3, 46, 47, 48, 49, 50, 3, 51, 52, 53, 3, 54, 3, 55, 56, 57, 58, 59, 3, 60, 61, 62, 3, 63, 64, 65, 66, 67, 3, 68, 69, 70, 71, 72, 73, 74, 3, 75, 76, 77, 3, 78, 3, 79, 80

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OFFSET

1,2

COMMENTS

The original name was: "Filter sequence for a(odd prime) = constant sequences", which stemmed from the fact that for all i, j, a(i) = a(j) => b(i) = b(j) for any sequence b that obtains a constant value for all odd primes A065091.

For example, we have for all i, j:

a(i) = a(j) => A305800(i) = A305800(j),

a(i) = a(j) => A007814(i) = A007814(j),

a(i) = a(j) => A305891(i) = A305891(j) => A291761(i) = A291761(j).

There are several filter sequences "above" this one (meaning that they have finer equivalence class partitioning), for example, we have, for all i, j:

[where odd primes are further distinguished by]

A305900(i) = A305900(j) => a(i) = a(j), [whether p = 3 or > 3]

A319350(i) = A319350(j) => a(i) = a(j), [A007733(p)]

A319704(i) = A319704(j) => a(i) = a(j), [p mod 4]

A319705(i) = A319705(j) => a(i) = a(j), [A286622(p)]

A331304(i) = A331304(j) => a(i) = a(j), [parity of A000720(p)]

A336855(i) = A336855(j) => a(i) = a(j). [distance to the next larger prime]

LINKS

Antti Karttunen, Table of n, a(n) for n = 1..100000

FORMULA

a(1) = 1, a(2) = 2; for n > 2, a(n) = 3 for odd primes, and a(n) = 2+n-A000720(n) for composite n.

For n > 2, a(n) = 1 + A305800(n).