A295300 -id:A295300 - 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

A305800 Filter sequence for a(prime) = constant sequences.

Original entry on oeis.org

1, 2, 2, 3, 2, 4, 2, 5, 6, 7, 2, 8, 2, 9, 10, 11, 2, 12, 2, 13, 14, 15, 2, 16, 17, 18, 19, 20, 2, 21, 2, 22, 23, 24, 25, 26, 2, 27, 28, 29, 2, 30, 2, 31, 32, 33, 2, 34, 35, 36, 37, 38, 2, 39, 40, 41, 42, 43, 2, 44, 2, 45, 46, 47, 48, 49, 2, 50, 51, 52, 2, 53, 2, 54, 55, 56, 57, 58, 2, 59, 60, 61, 2, 62, 63, 64, 65, 66, 2, 67, 68, 69, 70, 71, 72, 73, 2, 74, 75, 76, 2, 77, 2, 78, 79, 80, 2, 81, 2, 82, 83, 84, 2, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96

Offset: 1

Antti Karttunen, Jun 14 2018

nonn

Restricted growth sequence transform of A239968.
In the following, A stands for this sequence, A305800, and S -> T (where S and T are sequence A-numbers) indicates that for all i, j: S(i) = S(i) => T(i) = T(j).
For example, the following implications hold:
  A -> A300247 -> A305897 -> A077462 -> A101296,
  A -> A290110 -> A300250 -> A101296.

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

Cf. A000720, A239968.
Differs from A296073 for the first time at n=125, as a(125) = 96, while A296073(125) = 33.
Cf. also A305900, A305801, A295300, A289626 for other "upper level" filters.

Join[{1},Table[If[PrimeQ[n],2,1+n-PrimePi[n]],{n,2,150}]] (* Harvey P. Dale, Jul 12 2019 *)

A305800(n) = if(1==n,n,if(isprime(n),2,1+n-primepi(n)));

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

A324400 Lexicographically earliest sequence such that a(i) = a(j) => f(i) = f(j) for all i, j >= 1, where f(n) = -1 if n = 2^k and k > 0, and f(n) = n for all other numbers.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Mar 01 2019

nonn

In the following, A stands for this sequence, A324400, and S -> T (where S and T are sequence A-numbers) indicates that for all i, j >= 1: S(i) = S(i) => T(i) = T(j).
For example, the following chains of implications hold:
  A -> A286619 -> A005811,
and
  A -> A003602 -> A286622 -> A000120,
               -> A286378 -> A002487,
               -> A323889,
               -> A000593,
               -> A001227,
among many others.

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

Cf. A000523.
Cf. A000120, A000265, A000593, A001227, A002487, A003602, A005811, A209229, A286378, A286619, A286622, A294898, A297159, A318311, A323234, A323889, A323904, A324343, A324345 (some of the matching sequences).
Cf. also A103391, A295300, A305800, A305801, A324401.

A000523(n) = if(n<1, 0, #binary(n)-1);
A324400(n) = if(n<4,n,if(!bitand(n,n-1),2,1+n-A000523(n)));

If n <= 3, a(n) = n; and for n >= 4, if A209229(n) = 1, then a(n) = 2, otherwise a(n) = 1 + n - A000523(n).

A291751 Lexicographically earliest such sequence a that a(i) = a(j) => A003557(i) = A003557(j) and A048250(i) = A048250(j), for all i, j.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Sep 06 2017

nonn

Restricted growth sequence transform of A291750, which means that this is the lexicographically least sequence a, such that for all i, j: a(i) = a(j) <=> A291750(i) = A291750(j) <=> A003557(i) = A003557(j) and A048250(i) = A048250(j). That this is equal to the definition given in the title follows because any such lexicographically least sequence satisfying relation <=> is also the least sequence satisfying relation => with the same parameters.

Sigma (A000203) and psi (A001615) are functions of this sequence. See comments in A291750 for the reason. For example, to find the value of A001615(n) when we know just a(n), but without knowing n, let m be the least i for which a(i) = a(n); then A001615(n) = A003991(A291750(m)) = A003557(m) * A048250(m).

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

Cf. A001615, A003557, A048250, A291750, A291752, A294877, A295300, A295886, A295887, A295888, A319698, A322021.

Differs from A286603 for the first time at n = 25, where a(25) = 21, while A286603(25) = 14.

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) = n/factorback(factor(n)[, 1]); \\ From A003557
A048250(n) = if(n<1, 0, sumdiv(n, d, if(core(d)==d, d)));
A291750(n) = (1/2)*(2 + ((A003557(n)+A048250(n))^2) - A003557(n) - 3*A048250(n));
v291751 = rgs_transform(vector(65537,n,A291750(n)));
A291751(n) = v291751[n];

Name changed and comments added by Antti Karttunen, Nov 24 2018

A295887 Filter sequence combining A003557(n) and A173557(n); the restricted growth sequence transform of A291756.

Original entry on oeis.org

1, 1, 2, 3, 4, 2, 5, 6, 7, 4, 8, 9, 10, 5, 11, 12, 13, 7, 14, 15, 10, 8, 16, 17, 18, 10, 19, 20, 21, 11, 22, 23, 24, 13, 25, 26, 27, 14, 25, 28, 29, 10, 30, 31, 32, 16, 33, 34, 35, 18, 36, 37, 38, 19, 29, 39, 27, 21, 40, 41, 42, 22, 43, 44, 45, 24, 46, 47, 48, 25, 49, 50, 51, 27, 52, 53, 42, 25, 54, 55, 56, 29, 57, 37, 58, 30, 59, 60, 61, 32, 51, 62, 42, 33, 51

Offset: 1

Antti Karttunen, Dec 03 2017

nonn

First define function f(n) = (1/2)*(2 + ((A003557(n) + A173557(n))^2) - A003557(n) - 3*A173557(n)), or in short, f(n) = P(A003557(n),  A173557(n)), where P(n,k) is triangular table sequence A000027 used as an injective N x N -> N pairing function. Then apply the restricted growth sequence transform to the sequence f(1), f(2), f(3), ... See the example-section.
This is also the restricted growth sequence transform of sequence A291756, as A291756(n) = P(A003557(n), A000010(n)), where again P(n,k) is sequence A000027 used as a pairing function. Given either an ordered pair (A003557(n),A000010(n)) or (A003557(n),A173557(n)), the other one can be computed because A000010(n) = A003557(n)*A173557(n).
Note that the exact pairing function P used is not important, as long as it provides an injective mapping N x N -> N. So instead of Cantor's mapping we could as well used bit-interleaving A054238 (Morton code) to pack together A003557(n) and A173557(n), or equally, A000010(n) and A003557(n).

			The first ten terms of the sequence f(n) = (1/2)*(2 + ((A003557(n) + A173557(n))^2) - A003557(n) - 3*A173557(n)) are 1, 1, 2, 3, 7, 2, 16, 10, 9, 7. When we assign to each newly occurring term the least unused number k so far (starting by giving k=1 for the initial term, this k increases by one for each new distinct term produced by f(n) when n grows), and for each repeated term the same number it was given the previous time (equal to the number it was given for the first time), we obtain 1, 1, 2, 3, 4, 2, 5, 6, 7, 4, the first 10 terms of this sequence. Note how f(10) = 7 gets 4 because when seven occurred for the first time at f(5), it was the 4th distinct new number in that sequence.
This is also true for the sequence A291756 although there the terms are different: 1, 1, 2, 5, 7, 2, 16, 25, 31, 7.

Antti Karttunen, Table of n, a(n) for n = 1..65537
Eric Weisstein's World of Mathematics, Pairing Function
Wikipedia, Pairing Function

Cf. A000010, A003557, A054238, A173557, A291750, A291756, A295300, A295886.

allocatemem(2^30);
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; };
write_to_bfile(start_offset,vec,bfilename) = { for(n=1, length(vec), write(bfilename, (n+start_offset)-1, " ", vec[n])); }
A003557(n) = { my(f=factor(n)); for (i=1, #f~, f[i, 2] = max(0,f[i, 2]-1)); factorback(f); };
A173557(n) = my(f=factor(n)[, 1]); prod(k=1, #f, f[k]-1); \\ This function from Michel Marcus, Oct 31 2017
Anotsubmitted7(n) = (1/2)*(2 + ((A003557(n)+A173557(n))^2) - A003557(n) - 3*A173557(n));
write_to_bfile(1,rgs_transform(vector(up_to,n,Anotsubmitted7(n))),"b295887.txt");

A294877 Lexicographically earliest such sequence a that a(i) = a(j) => A003557(i) = A003557(j) and A046523(i) = A046523(j), for all i, j.

Original entry on oeis.org

1, 2, 2, 3, 2, 4, 2, 5, 6, 4, 2, 7, 2, 4, 4, 8, 2, 9, 2, 7, 4, 4, 2, 10, 11, 4, 12, 7, 2, 13, 2, 14, 4, 4, 4, 15, 2, 4, 4, 10, 2, 13, 2, 7, 9, 4, 2, 16, 17, 18, 4, 7, 2, 19, 4, 10, 4, 4, 2, 20, 2, 4, 9, 21, 4, 13, 2, 7, 4, 13, 2, 22, 2, 4, 18, 7, 4, 13, 2, 16, 23, 4, 2, 20, 4, 4, 4, 10, 2, 24, 4, 7, 4, 4, 4, 25, 2, 26, 9, 27, 2, 13, 2, 10, 13, 4, 2, 28, 2, 13

Offset: 1

Antti Karttunen, Nov 11 2017

nonn

Restricted growth sequence transform of A291757, which means that this is the lexicographically least sequence a, such that for all i, j: a(i) = a(j) <=> A291757(i) = A291757(j) <=> A003557(i) = A003557(j) and A046523(i) = A046523(j). That this is equal to the definition given in the title follows because any such lexicographically least sequence satisfying relation <=> is also the least sequence satisfying relation => with the same parameters.
Also the restricted growth sequence transform of A294876, Product_{d|n, d>1} prime(gcd(d,n/d)). (This was the original definition).
For all i, j:
  A295300(i) = A295300(j) => a(i) = a(j),
  A319347(i) = A319347(j) => a(i) = a(j),
  a(i) = a(j) => A055155(i) = A055155(j).

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

Cf. also A291751, A291757, A294876, A295300, A295888, A319347, A322021.

Cf. A000188, A055155, A294897, A295666, A322020 (a few of the matched sequences).

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; };
A294876(n) = { my(m=1); fordiv(n,d,if(d>1, m *= prime(gcd(d,n/d)))); m; };
v294877 = rgs_transform(vector(up_to,n,A294876(n)));
A294877(n) = v294877[n];

A003557(n) = n/factorback(factor(n)[, 1]); \\ From A003557
A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); };  \\ From A046523
v294877 = rgs_transform(vector(up_to,n,[A003557(n),A046523(n)]));
A294877(n) = v294877[n]; \\ Antti Karttunen, Nov 28 2018

Name changed and comments added by Antti Karttunen, Nov 28 2018

A295888 Filter combining prime signature of n (A101296) with Dedekind's psi (A001615).

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Dec 03 2017

nonn

Antti Karttunen, Table of n, a(n) for n = 1..65537
Eric Weisstein's World of Mathematics, Pairing Function
Wikipedia, Pairing Function

Cf. A001615, A046523, A101296, A291750, A295300, A295880, A295887, A296090.

allocatemem(2^30);
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; };
write_to_bfile(start_offset,vec,bfilename) = { for(n=1, length(vec), write(bfilename, (n+start_offset)-1, " ", vec[n])); }
A001615(n) = (n * sumdivmult(n, d, issquarefree(d)/d)); \\ This function from Charles R Greathouse IV, Sep 09 2014
A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); };  \\ This function from Charles R Greathouse IV, Aug 17 2011
Anotsubmitted8(n) = (1/2)*(2 + ((A046523(n)+A001615(n))^2) - A046523(n) - 3*A001615(n));
write_to_bfile(1,rgs_transform(vector(up_to,n,Anotsubmitted8(n))),"b295888.txt");

Restricted growth sequence transform of function f(n) = (1/2)*(2 + ((A046523(n) + A001615(n))^2) - A046523(n) - 3*A001615(n)), where values A046523(n) and A001615(n) are packed together to a(n) with the 2-argument form of A000027, also known as Cantor pairing-function.

A296090 Filter combining the sum of divisors (A000203) and prime-signature (A101296) of n; restricted growth sequence transform of A286360.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Dec 07 2017

nonn

For all i, j:
  a(i) = a(j) => A286034(i) = A286034(j).
  a(i) = a(j) => A295880(i) = A295880(j).

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

Cf. A000203, A101296, A046523, A286034, A286360, A295300, A295888, A296089.

Differs from related A295880 for the first time at n=135, where a(135) = 123, while A295880(135) = 104.

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; };
write_to_bfile(start_offset,vec,bfilename) = { for(n=1, length(vec), write(bfilename, (n+start_offset)-1, " ", vec[n])); }
A000203(n) = sigma(n);
A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); };  \\ This function from Charles R Greathouse IV, Aug 17 2011
A286360(n) = (1/2)*(2 + ((A046523(n)+A000203(n))^2) - A046523(n) - 3*A000203(n));
write_to_bfile(1,rgs_transform(vector(up_to,n,A286360(n))),"b296090.txt");

A322021 Lexicographically earliest such sequence a that a(i) = a(j) => A046523(i) = A046523(j) and A048250(i) = A048250(j), for all i, j.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Nov 29 2018

nonn

Restricted growth sequence transform of A291758, which means that this is the lexicographically least sequence a, such that for all i, j: a(i) = a(j) <=> A291758(i) = A291758(j) <=> A046523(i) = A046523(j) and A048250(i) = A048250(j). That this is equal to the definition given in the title follows because any such lexicographically least sequence satisfying relation <=> is also the least sequence satisfying relation => with the same parameters.
For all i, j:
  A295300(i) = A295300(j) => a(i) = a(j),
  a(i) = a(j) => A304411(i) = A304411(j),
  a(i) = a(j) => A304412(i) = A304412(j).

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

Cf. A046523, A048250, A291751, A291758, A294877, A295300, A322022.

Cf. also A304411, A304412.

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; };
A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); };  \\ From A046523
A048250(n) = if(n<1, 0, sumdiv(n, d, if(core(d)==d, d)));
v322021 = rgs_transform(vector(up_to, n, [A046523(n), A048250(n)]));
A322021(n) = v322021[n];

A326199 Lexicographically earliest sequence such that a(i) = a(j) => f(i) = f(j), where f(n) = [A003557(n), A046523(n), A048250(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, 10, 11, 3, 12, 3, 13, 14, 15, 3, 16, 17, 18, 19, 20, 3, 21, 3, 22, 23, 24, 23, 25, 3, 26, 27, 28, 3, 29, 3, 30, 31, 32, 3, 33, 34, 35, 32, 36, 3, 37, 32, 38, 39, 40, 3, 41, 3, 42, 43, 44, 45, 46, 3, 47, 42, 46, 3, 48, 3, 49, 50, 51, 42, 52, 3, 53, 54, 55, 3, 56, 57, 58, 59, 60, 3, 61, 62, 63, 64, 65, 59, 66, 3, 67, 68, 69, 3, 70, 3

Offset: 1

Antti Karttunen, Jul 13 2019

nonn

For all i, j:
  A295300(i) = A295300(j) => a(i) = a(j),
  A305801(i) = A305801(j) => a(i) = a(j),
  a(i) = a(j) => A294877(i) = A294877(j).

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

Cf. A003557, A046523, A048250, A294877, A295300, A305801.

Differs from A323401 for the first time at n = 382 where a(382) = 253, while A323401(382) = 140.

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) = n/factorback(factor(n)[, 1]); \\ From A003557
A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); };  \\ From A046523
A048250(n) = if(n<1, 0, sumdiv(n, d, if(core(d)==d, d)));
A291750(n) = (1/2)*(2 + ((A003557(n)+A048250(n))^2) - A003557(n) - 3*A048250(n));
Aux326199(n) = if((n>2)&&isprime(n),0,(1/2)*(2 + ((A046523(n) + A291750(n))^2) - A046523(n) - 3*A291750(n)));
v326199 = rgs_transform(vector(up_to,n,Aux326199(n)));
A326199(n) = v326199[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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A305800 Filter sequence for a(prime) = constant sequences.

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A324400 Lexicographically earliest sequence such that a(i) = a(j) => f(i) = f(j) for all i, j >= 1, where f(n) = -1 if n = 2^k and k > 0, and f(n) = n for all other numbers.

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A291751 Lexicographically earliest such sequence a that a(i) = a(j) => A003557(i) = A003557(j) and A048250(i) = A048250(j), for all i, j.

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A295887 Filter sequence combining A003557(n) and A173557(n); the restricted growth sequence transform of A291756.

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A294877 Lexicographically earliest such sequence a that a(i) = a(j) => A003557(i) = A003557(j) and A046523(i) = A046523(j), for all i, j.

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A295888 Filter combining prime signature of n (A101296) with Dedekind's psi (A001615).

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A296090 Filter combining the sum of divisors (A000203) and prime-signature (A101296) of n; restricted growth sequence transform of A286360.

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