A319701 -id:A319701 - 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

A323237 Lexicographically earliest sequence such that for all i, j, a(i) = a(j) => f(i) = f(j), where f(n) = A291756(n) for all n, except f(1) = -1 and for odd numbers n > 1, f(n) = 0.

Original entry on oeis.org

1, 2, 3, 4, 3, 5, 3, 6, 3, 7, 3, 8, 3, 9, 3, 10, 3, 11, 3, 12, 3, 13, 3, 14, 3, 15, 3, 16, 3, 17, 3, 18, 3, 19, 3, 20, 3, 21, 3, 22, 3, 15, 3, 23, 3, 24, 3, 25, 3, 26, 3, 27, 3, 28, 3, 29, 3, 30, 3, 31, 3, 32, 3, 33, 3, 34, 3, 35, 3, 36, 3, 37, 3, 38, 3, 39, 3, 36, 3, 40, 3, 41, 3, 27, 3, 42, 3, 43, 3, 44, 3, 45, 3, 46, 3, 47, 3, 48, 3, 49, 3, 50, 3, 51, 3

Offset: 1

Antti Karttunen, Jan 08 2019

nonn

For all i, j:

A319701(i) = A319701(j) => a(i) = a(j) => A319998(i) = A319998(j).

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

Cf. A003557, A173557, A291756, A295887, A319701, A319998, A322587, A323238, A323241.

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
Aux323237(n) = if(1==n,-1,if(n%2,0,(1/2)*(2 + ((A003557(n)+A173557(n))^2) - A003557(n) - 3*A173557(n))));
v323237 = rgs_transform(vector(up_to, n, Aux323237(n)));
A323237(n) = v323237[n];

A323238 Lexicographically earliest sequence such that for all i, j, a(i) = a(j) => f(i) = f(j), where f(n) = A291750(n) for all n, except for odd numbers n > 1, f(n) = 0.

Original entry on oeis.org

1, 2, 3, 4, 3, 5, 3, 6, 3, 7, 3, 8, 3, 9, 3, 10, 3, 11, 3, 12, 3, 13, 3, 14, 3, 15, 3, 16, 3, 17, 3, 18, 3, 19, 3, 20, 3, 21, 3, 22, 3, 23, 3, 24, 3, 17, 3, 25, 3, 26, 3, 27, 3, 28, 3, 29, 3, 30, 3, 31, 3, 23, 3, 32, 3, 33, 3, 34, 3, 33, 3, 35, 3, 36, 3, 37, 3, 38, 3, 39, 3, 40, 3, 41, 3, 42, 3, 43, 3, 44, 3, 31, 3, 33, 3, 45, 3, 46, 3, 47, 3, 48, 3, 49, 3

Offset: 1

Antti Karttunen, Jan 08 2019

nonn

For all i, j:
  A319701(i) = A319701(j) => a(i) = a(j),
  a(i) = a(j) => A146076(i) = A146076(j),
  a(i) = a(j) => A319697(i) = A319697(j).

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

Cf. A003557, A048250, A146076, A291750, A291751, A319698, A319697, A319701, A322588, A323237, A323241.

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]));
Aux323238(n) = if((n>1)&&(n%2),0,(1/2)*(2 + ((A003557(n)+A048250(n))^2) - A003557(n) - 3*A048250(n)));
v323238 = rgs_transform(vector(up_to, n, Aux323238(n)));
A323238(n) = v323238[n];

A323241 Lexicographically earliest sequence such that a(i) = a(j) => f(i) = f(j) for all i, j, where f(n<=2) = -n, f(n) = 0 if n is an odd number > 1, and f(n) = A300226(n) for even numbers >= 4.

Original entry on oeis.org

1, 2, 3, 4, 3, 5, 3, 6, 3, 5, 3, 7, 3, 5, 3, 8, 3, 9, 3, 7, 3, 5, 3, 10, 3, 5, 3, 7, 3, 11, 3, 12, 3, 5, 3, 13, 3, 5, 3, 10, 3, 11, 3, 7, 3, 5, 3, 14, 3, 15, 3, 7, 3, 16, 3, 10, 3, 5, 3, 17, 3, 5, 3, 18, 3, 11, 3, 7, 3, 19, 3, 20, 3, 5, 3, 7, 3, 11, 3, 14, 3, 5, 3, 17, 3, 5, 3, 10, 3, 21, 3, 7, 3, 5, 3, 22, 3, 23, 3, 24, 3, 11, 3, 10, 3

Offset: 1

Antti Karttunen, Jan 07 2019

nonn

For all i, j:
  A319701(i) = A319701(j) => a(i) = a(j),
  a(i) = a(j) => A007814(i) = A007814(j).
  a(i) = a(j) => A183063(i) = A183063(j).

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

Cf. A007814, A052126, A183063, A300226, A319701, A319988, A323235, A323237, A323238, A323242.

up_to = 10000;
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; };
A052126(n) = if(1==n, n, n/vecmax(factor(n)[, 1]));
A319988(n) = ((n>1)&&(factor(n)[omega(n),2]>1));
A323241aux(n) = if(n<=2,-n,if(n%2,0,[A052126(n), A319988(n)]));
v323241 = rgs_transform(vector(up_to,n,A323241aux(n)));
A323241(n) = v323241[n];

A319702 Filter sequence for sequences that are constant for all even terms >= 2.

Original entry on oeis.org

1, 2, 3, 2, 4, 2, 5, 2, 6, 2, 7, 2, 8, 2, 9, 2, 10, 2, 11, 2, 12, 2, 13, 2, 14, 2, 15, 2, 16, 2, 17, 2, 18, 2, 19, 2, 20, 2, 21, 2, 22, 2, 23, 2, 24, 2, 25, 2, 26, 2, 27, 2, 28, 2, 29, 2, 30, 2, 31, 2, 32, 2, 33, 2, 34, 2, 35, 2, 36, 2, 37, 2, 38, 2, 39, 2, 40, 2, 41, 2, 42, 2, 43, 2, 44, 2, 45, 2, 46, 2, 47, 2, 48, 2, 49, 2, 50, 2, 51, 2, 52, 2, 53, 2, 54, 2, 55, 2, 56, 2, 57, 2, 58, 2, 59, 2, 60, 2, 61, 2

Offset: 1

Antti Karttunen, Oct 02 2018

Restricted growth sequence transform of A141310.

For n > 2, a(n-1) is the number of occurrences of n in A319840. - Stefano Spezia, Apr 07 2023

Antti Karttunen, Table of n, a(n) for n = 1..100000
Index entries for linear recurrences with constant coefficients, signature (0,2,0,-1).

Cf. A141310, A319701, A319840.

A319702(n) = if(n<=2, n, if(!(n%2), 2, (n+3)/2));

a(1) = 1, and for n > 1, if n is even, a(n) = 2, otherwise a(n) = (n+3)/2.
From Stefano Spezia, Apr 07 2023: (Start)
O.g.f.: x*(1 + 2*x + x^2 - 2*x^3 - x^4)/((1 - x)^2*(1 + x)^2).
E.g.f.: ((4 + x)*cosh(x) + 3*sinh(x) - 2*(2 + x))/2. (End)

A323235 Lexicographically earliest sequence such that a(i) = a(j) => f(i) = f(j), where f(1) = 0, f(n) = -1 if n is an odd number > 1, and f(n) = A323234(n) for even numbers >= 4.

Original entry on oeis.org

1, 2, 3, 2, 3, 4, 3, 2, 3, 5, 3, 6, 3, 7, 3, 2, 3, 5, 3, 8, 3, 9, 3, 10, 3, 11, 3, 12, 3, 13, 3, 2, 3, 5, 3, 8, 3, 9, 3, 14, 3, 15, 3, 16, 3, 17, 3, 18, 3, 19, 3, 20, 3, 21, 3, 22, 3, 23, 3, 24, 3, 25, 3, 2, 3, 5, 3, 8, 3, 9, 3, 14, 3, 15, 3, 16, 3, 17, 3, 26, 3, 27, 3, 28, 3, 29, 3, 30, 3, 31, 3, 32, 3, 33, 3, 34, 3, 35, 3, 36, 3, 37, 3, 38, 3

Offset: 1

Antti Karttunen, Jan 08 2019

nonn

For all i, j:
  A319701(i) = A319701(j) => a(i) = a(j),
  A323234(i) = A323234(j) => a(i) = a(j).

Antti Karttunen, Table of n, a(n) for n = 1..65537
Index entries for sequences related to binary expansion of n

Cf. A053645, A079944, A319701, A323234.

Cf. also A323241 (somewhat analogous filter sequence for prime factorization).

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; };
A053644(n) = { my(k=1); while(k<=n, k<<=1); (k>>1); }; \\ From A053644
A053645(n) = (n-A053644(n));
A079944off0(n) = (1==binary(2+n)[2]);
A323235aux(n) = if(1==n,0,if(n%2,-1,[A053645(n), A079944off0(n-2)]));
v323235 = rgs_transform(vector(up_to,n,A323235aux(n)));
A323235(n) = v323235[n];

A323242 Lexicographically earliest sequence such that a(i) = a(j) => f(i) = f(j) for all i, j, where f(n<=2) = -n, f(n) = 0 if n is an even number > 2, and f(n) = A300226(n) for odd numbers >= 3.

Original entry on oeis.org

1, 2, 3, 4, 3, 4, 3, 4, 5, 4, 3, 4, 3, 4, 6, 4, 3, 4, 3, 4, 6, 4, 3, 4, 7, 4, 8, 4, 3, 4, 3, 4, 6, 4, 9, 4, 3, 4, 6, 4, 3, 4, 3, 4, 10, 4, 3, 4, 11, 4, 6, 4, 3, 4, 9, 4, 6, 4, 3, 4, 3, 4, 10, 4, 9, 4, 3, 4, 6, 4, 3, 4, 3, 4, 12, 4, 13, 4, 3, 4, 14, 4, 3, 4, 9, 4, 6, 4, 3, 4, 13, 4, 6, 4, 9, 4, 3, 4, 10, 4, 3, 4, 3, 4, 15

Offset: 1

Antti Karttunen, Jan 07 2019

nonn

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

Cf. A052126, A300226, A319701, A319988, A323241.

up_to = 10000;
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; };
A052126(n) = if(1==n, n, n/vecmax(factor(n)[, 1]));
A319988(n) = ((n>1)&&(factor(n)[omega(n),2]>1));
A323242aux(n) = if(n<=2,-n,if(!(n%2),0,[A052126(n), A319988(n)]));
v323242 = rgs_transform(vector(up_to,n,A323242aux(n)));
A323242(n) = v323242[n];

A328765 Lexicographically earliest infinite sequence such that a(i) = a(j) => f(i) = f(j), where f(n) = A328763(n), except for odd numbers > 1, f(n) = 0.

Original entry on oeis.org

1, 2, 3, 4, 5, 4, 3, 4, 5, 4, 6, 4, 7, 4, 6, 4, 8, 4, 9, 4, 10, 4, 10, 4, 11, 4, 10, 4, 12, 4, 7, 4, 5, 4, 9, 4, 9, 4, 6, 4, 10, 4, 13, 4, 10, 4, 10, 4, 14, 4, 12, 4, 12, 4, 15, 4, 12, 4, 16, 4, 13, 4, 6, 4, 8, 4, 14, 4, 10, 4, 10, 4, 17, 4, 12, 4, 12, 4, 18, 4, 12, 4, 19, 4, 20, 4, 21, 4, 19, 4, 17, 4, 10, 4, 12, 4, 14, 4, 12, 4, 12, 4, 22, 4, 12, 4

Offset: 0

Antti Karttunen, Oct 28 2019

nonn

For all i, j:
  A319701(i) = A319701(j) => a(i) = a(j),
  a(i) = a(j) => A328578(i) = A328578(j).

Antti Karttunen, Table of n, a(n) for n = 0..32768
Index entries for sequences related to primorial base

Cf. A276086, A319701, A328578, A328613, A328763, A328764.

up_to = 32768;
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; };
A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
A328613(n) = { my(m=1, p=2); while(n, m *= p^valuation(n,p); n = n\p; p = nextprime(1+p)); (m*p); };
A328763(n) = A328613(A276086(n));
Aux328765(n) = if((n>1)&&(n%2),0,A328763(n));
v328765 = rgs_transform(vector(1+up_to, n, Aux328765(n-1)));
A328765(n) = v328765[1+n];

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.

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A323237 Lexicographically earliest sequence such that for all i, j, a(i) = a(j) => f(i) = f(j), where f(n) = A291756(n) for all n, except f(1) = -1 and for odd numbers n > 1, f(n) = 0.

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A323238 Lexicographically earliest sequence such that for all i, j, a(i) = a(j) => f(i) = f(j), where f(n) = A291750(n) for all n, except for odd numbers n > 1, f(n) = 0.

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A323241 Lexicographically earliest sequence such that a(i) = a(j) => f(i) = f(j) for all i, j, where f(n<=2) = -n, f(n) = 0 if n is an odd number > 1, and f(n) = A300226(n) for even numbers >= 4.

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A319702 Filter sequence for sequences that are constant for all even terms >= 2.

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A323235 Lexicographically earliest sequence such that a(i) = a(j) => f(i) = f(j), where f(1) = 0, f(n) = -1 if n is an odd number > 1, and f(n) = A323234(n) for even numbers >= 4.

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A323242 Lexicographically earliest sequence such that a(i) = a(j) => f(i) = f(j) for all i, j, where f(n<=2) = -n, f(n) = 0 if n is an even number > 2, and f(n) = A300226(n) for odd numbers >= 3.

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A328765 Lexicographically earliest infinite sequence such that a(i) = a(j) => f(i) = f(j), where f(n) = A328763(n), except for odd numbers > 1, f(n) = 0.

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