A305801 -id:A305801 - cp's OEIS frontend

A353520 Lexicographically earliest infinite sequence such that a(i) = a(j) => A003415(i) = A003415(j), A003557(i) = A003557(j) and A053669(i) = A053669(j), for all i, j >= 1.

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, 29, 41, 42, 43, 3, 44, 3, 45, 46, 47, 48, 49, 3, 50, 51, 52, 3, 53, 3, 54, 55, 56, 48, 57, 3, 58, 59, 60, 3, 61, 42, 62, 63, 64, 3, 65, 38

Offset: 1

Antti Karttunen, Apr 25 2022

nonn

Restricted growth sequence transform of the triplet [A003415(n), A003557(n), A053669(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) => A344025(i) = A344025(j).

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

Cf. A003415, A003557, A007814, A053669.

Cf. also A305801, A328470, A344025, A351260.

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; };
A003415(n) = {my(fac); if(n<1, 0, fac=factor(n); sum(i=1, matsize(fac)[1], n*fac[i, 2]/fac[i, 1]))}; \\ From A003415
A003557(n) = (n/factorback(factorint(n)[, 1]));
A053669(n) = forprime(p=2, , if(n%p, return(p))); \\ From A053669
Aux353520(n) = [A003415(n), A003557(n), A053669(n)];
v353520 = rgs_transform(vector(up_to,n,Aux353520(n)));
A353520(n) = v353520[n];

A319346 Filter sequence combining the sum of proper divisors (A001065) with the 2-adic valuation of n (A007814).

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, 20, 26, 3, 27, 28, 29, 3, 30, 3, 31, 32, 33, 3, 34, 35, 36, 37, 38, 3, 39, 28, 40, 41, 42, 3, 43, 3, 44, 45, 46, 47, 48, 3, 49, 50, 51, 3, 52, 3, 53, 54, 55, 47, 56, 3, 57, 58, 59, 3, 60, 41, 61, 32, 62, 3, 63, 37, 64, 65, 66, 67, 68, 3, 69

Offset: 1

Antti Karttunen, Sep 24 2018

nonn

Restricted growth sequence transform of ordered pair [A001065(n), A007814(n)].

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

Cf. A001065, A007814, A305801, A319338.

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; };
A001065(n) = (sigma(n)-n);
A007814(n) = valuation(n,2);
v319346 = rgs_transform(vector(up_to,n,[A001065(n),A007814(n)]));
A319346(n) = v319346[n];

A322809 Lexicographically earliest such sequence a that a(i) = a(j) => f(i) = f(j) for all i, j, where f(n) = -1 if n is an odd prime, and f(n) = floor(n/2) for all other numbers.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Dec 26 2018

nonn

This sequence is a restricted growth sequence transform of a function f which is defined as f(n) = A004526(n), unless n is an odd prime, in which case f(n) = -1, which is a constant not in range of A004526. See the Crossrefs section for a list of similar sequences.
For all i, j:
  A305801(i) = A305801(j) => a(i) = a(j),
  a(i) = a(j) => A039636(i) = A039636(j).
For all i, j: a(i) = a(j) <=> A323161(i+1) = A323161(j+1).
The shifted version of this filter, A323161, has a remarkable ability to find many sequences related to primes and prime chains. - Antti Karttunen, Jan 06 2019

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

Cf. A004526, A039636, A305801, A323161.
A list of few similarly constructed sequences follows, where each sequence is an rgs-transform of such function f, for which the value of f(n) is the n-th term of the sequence whose A-number follows after a parenthesis, unless n is of the form ..., in which case f(n) is given a constant value outside of the range of that sequence:
  A322809 (A004526, unless an odd prime) [This sequence],
  A322589 (A007913, unless an odd prime),
  A322591 (A007947, unless an odd prime),
  A322805 (A252463, unless an odd prime),
  A323082 (A300840, unless an odd prime),
  A322822 (A300840, unless n > 2 and a Fermi-Dirac prime, A050376),
  A322988 (A322990, unless a prime power > 2),
  A323078 (A097246, unless an odd prime),
  A322808 (A097246, unless a squarefree number > 2),
  A322816 (A048675, unless an odd prime),
  A322807 (A285330, unless an odd prime),
  A322814 (A286621, unless an odd prime),
  A322824 (A242424, unless an odd prime),
  A322973 (A006370, unless an odd prime),
  A322974 (A049820, unless n > 1 and n is in A046642),
  A323079 (A060681, unless an odd prime),
  A322587 (A295887, unless an odd prime),
  A322588 (A291751, unless an odd prime),
  A322592 (A289625, unless an odd prime),
  A323369 (A323368, unless an odd prime),
  A323371 (A295886, unless an odd prime),
  A323374 (A323373, unless an odd prime),
  A323401 (A323372, unless an odd prime),
  A323405 (A323404, unless an odd prime).

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; };
A322809aux(n) = if((n>2)&&isprime(n),-1,(n>>1));
v322809 = rgs_transform(vector(up_to,n,A322809aux(n)));
A322809(n) = v322809[n];

a(n) = A323161(n+1) - 1.

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];

A323405 Lexicographically earliest sequence such that for all i, j, a(i) = a(j) => f(i) = f(j) where f(n) = [A003557(n), A023900(n), A063994(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, 58, 3, 59, 60, 61, 3, 62, 63, 64, 65, 66, 3, 67, 68, 69, 57, 70, 71, 72, 3, 73, 74, 75, 3

Offset: 1

Antti Karttunen, Jan 13 2019

nonn

For all i, j:
  A305801(i) = A305801(j) => a(i) = a(j),
  a(i) = a(j) => A323371(i) = A323371(j),
  a(i) = a(j) => A247074(i) = A247074(j).

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

Cf. A000010, A003557, A023900, A063994, A247074, A305801, A323371, A323404.
Differs from A323370 for the first time at n=78, where a(78) = 58, while A323370(78) = 52.
Cf. also A323374.

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
A063994(n) = { my(f=factor(n)[, 1]); prod(i=1, #f, gcd(f[i]-1, n-1)); }; \\ From A063994
Aux323405(n) = if((n>2)&&isprime(n),0,[A003557(n), A023900(n), A063994(n)]);
v323405 = rgs_transform(vector(up_to, n, Aux323405(n)));
A323405(n) = v323405[n];

A322816 Lexicographically earliest sequence such that a(i) = a(j) => f(i) = f(j) for all i, j, where f(n) = A048675(n) for all other numbers, except f(n) = -1 if n is an odd prime.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Dec 27 2018

nonn

For all i, j > 1:
  A323078(i) = A323078(j) => a(i) = a(j),
  a(i) = a(j) => A322812(i) = A322812(j),
  a(i) = a(j) => A322869(i) = A322869(j).

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

Cf. A048675, A305801, A322812, A322815, A322869, A323078.

up_to = 4096;
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; };
A048675(n) = my(f = factor(n)); sum(k=1, #f~, f[k, 2]*2^primepi(f[k, 1]))/2; \\ From A048675
A322816aux(n) = if((n>2)&&isprime(n),-1,A048675(n));
v322816 = rgs_transform(vector(up_to,n,A322816aux(n)));
A322816(n) = v322816[n];

A323367 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) = A323366(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, 26, 27, 3, 28, 26, 29, 3, 19, 3, 30, 31, 32, 3, 33, 34, 35, 36, 37, 3, 38, 39, 40, 41, 42, 3, 43, 3, 44, 45, 46, 47, 48, 3, 49, 50, 51, 3, 52, 3, 53, 54, 55, 56, 51, 3, 57, 58, 59, 3, 37, 60, 61, 62, 63, 3, 64, 65, 66, 56, 67, 65, 68, 3, 69, 70, 71, 3, 72, 3, 73, 47

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), A173557(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) => A322587(i) = A322587(j).
  a(i) = a(j) => A323237(i) = A323237(j).

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

Cf. A000035, A003557, A173557, A291756, A305801, A322587, A323237, A323366.

Cf. also A323369.

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
Aux323367(n) = if((n>2)&&isprime(n),0,[(n%2), A003557(n), A173557(n)]);
v323367 = rgs_transform(vector(up_to, n, Aux323367(n)));
A323367(n) = v323367[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];

A323374 Lexicographically earliest sequence such that for all i, j, a(i) = a(j) => f(i) = f(j) where f(n) = A323373(n) for all other numbers, except f(p) = -(p mod 2) for primes p.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Jan 13 2019

nonn

For all i, j:
  A305801(i) = A305801(j) => a(i) = a(j),
  a(i) = a(j) => A039651(i) = A039651(j).

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

Cf. A039651, A049559, A160595, A323373.

Cf. also A322587, A322592, A323405.

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; };
A049559(n) = gcd(eulerphi(n), n-1);
A160595(n) = if(1==n, n, numerator(eulerphi(n)/(n-1)));
Aux323374(n) = if(isprime(n),-(n%2),[A049559(n), A160595(n)]);
v323374 = rgs_transform(vector(up_to, n, Aux323374(n)));
A323374(n) = v323374[n];

A329608 Lexicographically earliest infinite sequence such that a(i) = a(j) => f(i) = f(j), where f(n) = A329605(n) for all other n, except for odd primes p, f(p) = 0.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Nov 18 2019

nonn

For all i, j:

A305801(i) = A305801(j) => a(i) = a(j) => A329614(i) = A329614(j).

Cf. A000005, A108951, A305801, A329605, A329606, A329614.

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; };
A034386(n) = prod(i=1, primepi(n), prime(i));
A108951(n) = { my(f=factor(n)); prod(i=1, #f~, A034386(f[i, 1])^f[i, 2]) };  \\ From A108951
A329605(n) = numdiv(A108951(n));
Aux329608(n) = if((n%2)&&isprime(n),0,A329605(n));
v329608 = rgs_transform(vector(up_to, n, Aux329608(n)));
A329608(n) = v329608[n];

cp's OEIS Frontend

A353520 Lexicographically earliest infinite sequence such that a(i) = a(j) => A003415(i) = A003415(j), A003557(i) = A003557(j) and A053669(i) = A053669(j), for all i, j >= 1.

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A319346 Filter sequence combining the sum of proper divisors (A001065) with the 2-adic valuation of n (A007814).

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A322809 Lexicographically earliest such sequence a that a(i) = a(j) => f(i) = f(j) for all i, j, where f(n) = -1 if n is an odd prime, and f(n) = floor(n/2) for all other numbers.

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Formula

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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A323405 Lexicographically earliest sequence such that for all i, j, a(i) = a(j) => f(i) = f(j) where f(n) = [A003557(n), A023900(n), A063994(n)] for all other numbers, except f(n) = 0 for odd primes.

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A322816 Lexicographically earliest sequence such that a(i) = a(j) => f(i) = f(j) for all i, j, where f(n) = A048675(n) for all other numbers, except f(n) = -1 if n is an odd prime.

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A323367 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) = A323366(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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A323374 Lexicographically earliest sequence such that for all i, j, a(i) = a(j) => f(i) = f(j) where f(n) = A323373(n) for all other numbers, except f(p) = -(p mod 2) for primes p.

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