A319350 -id:A319350 - cp's OEIS frontend

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

Antti Karttunen, Jun 14 2018

nonn

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]

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

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).

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

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

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).

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

A319704 Filter sequence which records for primes their residue modulo 4, and for all other numbers assigns a unique number.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Sep 26 2018

nonn

Restricted growth sequence transform of function f defined as f(n) = A010873(n) when n is a prime, otherwise -n.
For all i, j:
  a(i) = a(j) => A010873(i) = A010873(j),
  a(i) = a(j) => A305801(i) = A305801(j),
  a(i) = a(j) => A319714(i) = A319714(j).

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

Cf. A010873, A305801, A319714.
Cf. A002145 (positions of 3's), A002144 (positions of 5's).
Cf. also A319350, A319705, A319706.

up_to = 100000;
rgs_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(om,invec[i]), my(pp = mapget(om, invec[i])); outvec[i] = outvec[pp] , mapput(om,invec[i],i); outvec[i] = u; u++ )); outvec; };
A319704aux(n) = if(isprime(n),-(n%4),n);
v319704 = rgs_transform(vector(up_to,n,A319704aux(n)));
A319704(n) = v319704[n];

A319351 Filter sequence which records the number of cyclotomic cosets of 2 mod p^k for powers of odd primes p, and for any other number assigns a unique number.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Sep 26 2018

nonn

All prime powers p^k, k >= 1, are allotted to distinct equivalence classes according to the number of cyclotomic cosets of 2 mod p^k, while all other numbers occur in singular equivalence classes of their own.
Restricted growth sequence transform of function f defined as f(n) = A006694((n-1)/2) when n is an odd prime power > 1, otherwise -n.
For all i, j: a(i) = a(j) => A305976(i) = A305976(j). (See also A305975).

			a(7) = a(9) = a(17) = a(23) = a(25) = a(41) = ... because n = 7, 9, 17, 23, 25, 41, ... are such powers of odd primes for which A006694((n-1)/2) = 4.

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

Cf. A001917, A006694, A286573, A305801, A319350.

up_to = 100000;
rgs_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(om,invec[i]), my(pp = mapget(om, invec[i])); outvec[i] = outvec[pp] , mapput(om,invec[i],i); outvec[i] = u; u++ )); outvec; };
A006694(n) = (sumdiv(2*n+1, d, eulerphi(d)/znorder(Mod(2, d))) - 1); \\ From A006694
A319351aux(n) = if((n<=2)||!(n%2)||!isprimepower(n),n,-(A006694((n-1)/2)));
v319351 = rgs_transform(vector(up_to,n,A319351aux(n)));
A319351(n) = v319351[n];

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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.

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A319704 Filter sequence which records for primes their residue modulo 4, and for all other numbers assigns a unique number.

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A319351 Filter sequence which records the number of cyclotomic cosets of 2 mod p^k for powers of odd primes p, and for any other number assigns a unique number.

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