A323367 -id:A323367 - 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

A323369 Lexicographically earliest such sequence a that for all i, j, a(i) = a(j) => f(i) = f(j), where f(n) = 0 for odd primes, and f(n) = A323368(n) for any other number.

Original entry on oeis.org

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, 24, 26, 3, 27, 28, 29, 3, 30, 3, 31, 32, 22, 3, 33, 34, 35, 36, 37, 3, 38, 36, 39, 40, 41, 3, 42, 3, 30, 43, 44, 45, 46, 3, 47, 48, 46, 3, 49, 3, 50, 51, 52, 48, 53, 3, 54, 55, 56, 3, 57, 58, 59, 60, 61, 3, 62, 63, 42, 64, 46, 60, 65, 3, 66, 67, 68, 3, 69, 3

Offset: 1

Antti Karttunen, Jan 12 2019

nonn

Restricted growth sequence transform of function f, where f(n) = 0 for odd primes, and for any other number, f(n) = [A000035(n), A003557(n), A048250(n)].
For all i, j:
  A305801(i) = A305801(j) => a(i) = a(j),
  a(i) = a(j) => A007814(i) = A007814(j),
  a(i) = a(j) => A322588(i) = A322588(j),
  a(i) = a(j) => A323238(i) = A323238(j).

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A000035, A003557, A048250, A291750, A291751, A305801, A322588, A323238, A323368.

Cf. also A323367.

up_to = 65537;
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; };
A003557(n) = { my(f=factor(n)); for(i=1, #f~, f[i, 2] = f[i, 2]-1); factorback(f); };
A048250(n) = factorback(apply(p -> p+1,factor(n)[,1]));
Aux323369(n) = if((n>2)&&isprime(n),0,[(n%2), A003557(n), A048250(n)]);
v323369 = rgs_transform(vector(up_to, n, Aux323369(n)));
A323369(n) = v323369[n];

A323370 Lexicographically earliest sequence such that for all i, j, a(i) = a(j) => f(i) = f(j) where f(n) = [A000035(n), A003557(n), A023900(n)] for all other numbers, except f(n) = 0 for odd primes.

Original entry on oeis.org

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, 26, 29, 3, 30, 3, 31, 32, 33, 3, 34, 35, 36, 37, 38, 3, 39, 40, 41, 42, 43, 3, 44, 3, 45, 46, 47, 48, 49, 3, 50, 51, 52, 3, 53, 3, 54, 55, 56, 57, 52, 3, 58, 59, 60, 3, 61, 62, 63, 64, 65, 3, 66, 67, 68, 57, 69, 67, 70, 3, 71, 72, 73, 3, 74, 3, 75, 76

Offset: 1

Antti Karttunen, Jan 13 2019

nonn

Restricted growth sequence transform of function f, defined as f(n) = 0 when n is an odd prime, and f(n) = [A000035(n), A003557(n), A023900(n)] for all other numbers.
For all i, j:
  A305801(i) = A305801(j) => a(i) = a(j),
  a(i) = a(j) => A323367(i) = A323367(j),
  a(i) = a(j) => A323371(i) = A323371(j).

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A000035, A003557, A023900, A092248, A295886, A305801, A319340, A322587, A323367, A323371.

Differs from A323405 for the first time at n=78, where a(78) = 52, while A323405(78) = 58.

up_to = 65537;
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; };
A003557(n) = { my(f=factor(n)); for (i=1, #f~, f[i, 2] = max(0,f[i, 2]-1)); factorback(f); };
A023900(n) = sumdivmult(n, d, d*moebius(d)); \\ From A023900
Aux323370(n) = if((n>2)&&isprime(n),0,[(n%2), A003557(n), A023900(n)]);
v323370 = rgs_transform(vector(up_to, n, Aux323370(n)));
A323370(n) = v323370[n];

A323366 Lexicographically earliest sequence such that for all i, j, a(i) = a(j) => A000035(i) = A000035(j) and A003557(i) = A003557(j) and A173557(i) = A173557(j).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 13, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 34, 38, 39, 25, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 39, 52, 36, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 56, 65, 72, 73, 74, 75, 76, 49, 77, 78, 79, 80, 81, 82, 68, 83, 56, 84, 68, 85, 86, 87, 88, 89, 90, 91, 92, 93, 60

Offset: 1

Antti Karttunen, Jan 12 2019

nonn

For all i, j:
  a(i) = a(j) => A007814(i) = A007814(j),
  a(i) = a(j) => A295887(i) = A295887(j),
  a(i) = a(j) => A323237(i) = A323237(j).

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A000035, A003557, A007814, A173557, A291756, A295887, A323237, A323367.

Cf. also A323368.

up_to = 65537;
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; };
A003557(n) = { my(f=factor(n)); for(i=1, #f~, f[i, 2] = f[i, 2]-1); factorback(f); };
A173557(n) = my(f=factor(n)[, 1]); prod(k=1, #f, f[k]-1); \\ From A173557
v323366 = rgs_transform(vector(up_to, n, [(n%2), A003557(n), A173557(n)]));
A323366(n) = v323366[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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A323369 Lexicographically earliest such sequence a that for all i, j, a(i) = a(j) => f(i) = f(j), where f(n) = 0 for odd primes, and f(n) = A323368(n) for any other number.

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A323370 Lexicographically earliest sequence such that for all i, j, a(i) = a(j) => f(i) = f(j) where f(n) = [A000035(n), A003557(n), A023900(n)] for all other numbers, except f(n) = 0 for odd primes.

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A323366 Lexicographically earliest sequence such that for all i, j, a(i) = a(j) => A000035(i) = A000035(j) and A003557(i) = A003557(j) and A173557(i) = A173557(j).

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