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

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).

Mathematica

Array[If[# <= 2, #, If[PrimeQ[#], 3, 2 + # - PrimePi[#]]] &, 105] (* Michael De Vlieger, Oct 18 2021 *)

Prog

(PARI) A305801(n) = if(n<=2, n, if(isprime(n), 3, 2+n-primepi(n)));

Crossrefs

Cf. A000720, A065091, A305800, A305900.

Cf. A305900, A319350, A319704, A319705, A331304, A336855 (sequences with finer equivalence class partitioning).

Cf. A305800, A305890, A305891, A305896, A318500, A318888, A319346, A319347, A319349, A319701, A322591, A322809, A322810, A323078, A323367, A323082, A323369, A323370, A323371, A323374, A323400, A324401, A326199, A326201, A326203, A326203, A328470, A329608, A331174, A331730, A331301 (sequences with coarser equivalence class partitioning).

Cf. also A003602, A103391, A295300, A305795, A324400, A331300, A336460 (for similar constructions or similarly useful sequences).

Sequence in context: A361021 A322973 A305890 * A075850 A327565 A054437

Adjacent sequences: A305798 A305799 A305800 * A305802 A305803 A305804

Keyword

nonn

Author

Antti Karttunen, Jun 14 2018

Extensions

Name changed and Comment section rewritten by Antti Karttunen, Oct 17 2021

Status

approved