A324400 -id:A324400 - cp's OEIS frontend

A336156 Lexicographically earliest infinite sequence such that a(i) = a(j) => A007814(1+i) = A007814(1+j) and A336158(i) = A336158(j), for all i, j >= 1.

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

Offset: 1

Antti Karttunen, Jul 11 2020

nonn

Restricted growth sequence transform of the ordered pair [A007814(1+n), A336158(n)]. Note that A007814(1+n) gives the number of trailing 1-bits in the binary expansion of n.

For all i, j: A324400(i) = A324400(j) => a(i) = a(j).

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

Cf. A000265, A007814, A046523, A336158.

Cf. also A324400, A336152, A336153, A336154, A336155.

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; };
A000265(n) = (n>>valuation(n,2));
A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); };  \\ From A046523
A336158(n) = A046523(A000265(n));
A007814(n) = valuation(n,2);
Aux336156(n) = [A007814(1+n), A336158(n)];
v336156 = rgs_transform(vector(up_to, n, Aux336156(n)));
A336156(n) = v336156[n];

A351461 Lexicographically earliest infinite sequence such that a(i) = a(j) => A206787(i) = A206787(j) and A336651(i) = A336651(j) for all i, j >= 1.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Feb 11 2022

nonn

Restricted growth sequence transform of the ordered pair [A206787(n), A336651(n)], or equally, of sequence b(n) = A291750(A000265(n)).
For all i, j >= 1:
  A003602(i) = A003602(j) => A351040(i) = A351040(j) => a(i) = a(j),
  A324400(i) = A324400(j) => A351460(i) = A351460(j) => a(i) = a(j),
  a(i) = a(j) => A000593(i) = A000593(j),
  a(i) = a(j) => A347385(i) = A347385(j),
  a(i) = a(j) => A351037(i) = A351037(j) => A347240(i) = A347240(j).
From Antti Karttunen, Nov 23 2023: (Start)
Conjectured to be equal to the lexicographically earliest infinite sequence such that b(i) = b(j) => A000593(i) = A000593(j) and A336467(i) = A336467(j) for all i, j >= 1. In any case, a(i) = a(j) => b(i) = b(j) for all i, j >= 1 [because both A000593(n) and A336467(n) can be computed from the values of A206787(n) and A336651(n)], but whether the implication holds to the opposite direction is still open. Empirically this has been checked up to n = 2^22. See also comment in A351040.
(End)

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

Cf. A206787, A336651.
Cf. also A000593, A003602, A291750, A291751, A324400, A336467, A347240, A347385, A351040, A351460.
Differs from A351037 for the first time at n=103, where a(103) = 42 while A351037(103) = 27.

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; };
A206787(n) = sumdiv(n, d, d*(d % 2)*issquarefree(d)); \\ From A206787
A336651(n) = { my(f=factor(n)); prod(i=1, #f~, if(2==f[i,1],1,f[i,1]^(f[i,2]-1))); };
Aux351461(n) = [A206787(n), A336651(n)];
v351461 = rgs_transform(vector(up_to, n, Aux351461(n)));
A351461(n) = v351461[n];

A336146 Lexicographically earliest infinite sequence such that a(i) = a(j) => A000035(i) = A000035(j) and A000265(i) = A000265(j), for all i, j >= 1.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Jul 12 2020

nonn

Restricted growth sequence transform of the ordered pair [A000035(n), A000265(n)] (parity and the odd part of n), or equally, of the ordered pair [A000265(n), A278221(n)].
For all i, j:
  A324400(i) = A324400(j) => a(i) = a(j),
  a(i) = a(j) => A336126(i) = A336126(j),
  a(i) = a(j) => A336147(i) = A336147(j),
  a(i) = a(j) => A336148(i) = A336148(j),
  a(i) = a(j) => A336149(i) = A336149(j).

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

Cf. A000035, A000265, A046523, A064989, A122111, A278221.

Cf. also A286621, A324400, A336126, A336147, A336148, A336149.

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; };
A000035(n) = (n%2);
A000265(n) = (n>>valuation(n,2));
Aux336146(n) = [A000035(n), A000265(n)];
v336146 = rgs_transform(vector(up_to, n, Aux336146(n)));
A336146(n) = v336146[n];

A336149 Lexicographically earliest infinite sequence such that a(i) = a(j) => A278221(i) = A278221(j) and A278222(i) = A278222(j), for all i, j >= 1.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Jul 12 2020

nonn

Restricted growth sequence transform of the ordered pair [A278221(n), A278222(n)], i.e., of the ordered pair [A046523(A122111(n)), A046523(A005940(1+n))].

For all i, j: A336146(i) = A336146(j) => a(i) = a(j) => A035531(i) = A035531(j).

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

Cf. A005940, A046523, A064989, A122111, A278221, A278222.

Cf. also A035531, A286621, A286622, A324400, A336146, A336148, A336159.

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; };
A005940(n) = { my(p=2, t=1); n--; until(!n\=2, if((n%2), (t*=p), p=nextprime(p+1))); (t); };
A064989(n) = {my(f); f = factor(n); if((n>1 && f[1,1]==2), f[1,2] = 0); for (i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f)};
A122111(n) = if(1==n,n,prime(bigomega(n))*A122111(A064989(n)));
A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); }; \\ From A046523
A278221(n) = A046523(A122111(n));
A278222(n) = A046523(A005940(1+n));
Aux336149(n) = [A278221(n),A278222(n)];
v336149 = rgs_transform(vector(up_to, n, Aux336149(n)));
A336149(n) = v336149[n];

A336160 Lexicographically earliest infinite sequence such that a(i) = a(j) => A335915(i) = A335915(j) and A336158(i) = A336158(j), for all i, j >= 1.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Jul 11 2020

nonn

Restricted growth sequence transform of the ordered pair [A335915(n), A336158(n)].

For all i, j: A324400(i) = A324400(j) => a(i) = a(j) => A336161(i) = A336161(j).

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

Cf. A000265, A046523, A324400, A335915, A336158, A336159, A336161, A336162.

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; };
A000265(n) = (n>>valuation(n,2));
A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); };  \\ From A046523
A336158(n) = A046523(A000265(n));
A335915(n) = { my(f=factor(n)); prod(k=1,#f~,if(2==f[k,1],1,(A000265(f[k,1]-1)*A000265(f[k,1]+1))^f[k,2])); };
Aux336160(n) = [A335915(n), A336158(n)];
v336160 = rgs_transform(vector(up_to, n, Aux336160(n)));
A336160(n) = v336160[n];

A336460 Lexicographically earliest infinite sequence such that a(i) = a(j) => f(i) = f(j), where f(n) = [A278222(n), A336158(n), A336466(n)], for all i, j >= 1.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Jul 24 2020

nonn

Restricted growth sequence transform of the ordered triple [A278222(n), A336158(n), A336466(n)].
For all i, j:
  A324400(i) = A324400(j) => A003602(i) = A003602(j) => a(i) = a(j),
  a(i) = a(j) => A336159(i) = A336159(j),
  a(i) = a(j) => A336470(i) = A336470(j) => A336471(i) = A336471(j),
  a(i) = a(j) => A336472(i) = A336472(j),
  a(i) = a(j) => A336473(i) = A336473(j).

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

Cf. A278222, A336158, A336466.

Cf. A003602, A324400, A336159, A336470, A336471, A336472, A336473.

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; };
A000265(n) = (n>>valuation(n,2));
A005940(n) = { my(p=2, t=1); n--; until(!n\=2, if((n%2), (t*=p), p=nextprime(p+1))); t };
A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); };  \\ From A046523
A278222(n) = A046523(A005940(1+n));
A336158(n) = A046523(A000265(n));
A336466(n) = { my(f=factor(n)); prod(k=1,#f~,if(2==f[k,1],1,(A000265(f[k,1]-1))^f[k,2])); };
Aux336460(n) = [A278222(n), A336158(n), A336466(n)];
v336460 = rgs_transform(vector(up_to, n, Aux336460(n)));
A336460(n) = v336460[n];

A351460 Lexicographically earliest infinite sequence such that a(i) = a(j) => A006530(i) = A006530(j), A206787(i) = A206787(j) and A336651(i) = A336651(j) for all i, j >= 1.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Feb 11 2022

nonn

Restricted growth sequence transform of the ordered triplet [A006530(n), A206787(n), A336651(n)].
For all i, j >= 1:
  A324400(i) = A324400(j) => a(i) = a(j),
  a(i) = a(j) => A347241(i) = A347241(j),
  a(i) = a(j) => A351461(i) = A351461(j) => A347240(i) = A347240(j).

			a(429) = a(455) because 429 = 3*11*13 and 455 = 5*7*13, so they have equal largest prime factor (A006530), and they also agree on A206787(429) = A206787(455) = 672 and on A336651(429) = A336651(455) = 1 (because both are squarefree), therefore they get equal value (which is 216) allotted to them by the restricted growth sequence transform. - _Antti Karttunen_, Feb 14 2022

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

Cf. A006530, A206787, A336651, A347241, A351461.
Cf. also A324400, A351452.
Differs from A351454 for the first time at n=121, where a(121) = 62, while A351454(121) = 51.
Differs from A103391(1+n) for the first time after n=1 at n=455, where a(455) = 216, while A103391(456) = 229.

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; };
A006530(n) = if(1==n, n, my(f=factor(n)); f[#f~, 1]);
A206787(n) = sumdiv(n, d, d*(d % 2)*issquarefree(d)); \\ From A206787
A336651(n) = { my(f=factor(n)); prod(i=1, #f~, if(2==f[i,1],1,f[i,1]^(f[i,2]-1))); };
Aux351460(n) = [A006530(n), A206787(n), A336651(n)];
v351460 = rgs_transform(vector(up_to, n, Aux351460(n)));
A351460(n) = v351460[n];

A324530 Lexicographically earliest sequence such that a(i) = a(j) => f(i) = f(j), where f(n) = [A033879(n), A318458(n)] for all other numbers, except f(1) = -1.

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, 9, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 2, 16, 28, 19, 29, 30, 31, 19, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 2, 39, 56, 57, 58, 35, 59, 60, 61, 62, 63, 64, 65, 51, 66, 67, 68, 41, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 51, 80

Offset: 1

Antti Karttunen, Mar 05 2019

nonn

Cf. A033879, A318458, A324389, A324400, A324531.

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; };
A033879(n) = (n+n-sigma(n));
A318458(n) = bitand(n,sigma(n)-n);
Aux324530(n) = if(1==n,-1,[A033879(n), A318458(n)]);
v324530 = rgs_transform(vector(up_to,n,Aux324530(n)));
A324530(n) = v324530[n];

a(2^n) = 2 for all n >= 1.

A331743 Lexicographically earliest infinite sequence such that a(i) = a(j) => A002487(i) = A002487(j) and A323901(i) = A323901(j) for all i, j.

Original entry on oeis.org

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

Offset: 0

Antti Karttunen, Feb 05 2020

nonn

Restricted growth sequence transform of the ordered pair [A002487(n), A002487(A163511(n))].
For all i, j:
  a(i) = a(j) => A331748(i) = A331748(j),
  a(i) = a(j) => A331749(i) = A331749(j).

Cf. A002487, A163511, A323901, A324400, A331742, A331744, A331748, A331749.
Differs from A331745 for the first time at n=77, where a(77) = 40, while A331745(77) = 24.
Differs from A103391(1+n) for the first time at n=191, where a(191) = 23, while A103391(192) = 97.

\\ Needs also code from A323901.
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; };
Aux331743(n) = [A002487(n), A323901(n)];
v331743 = rgs_transform(vector(1+up_to, n, Aux331743(n-1)));
A331743(n) = v331743[1+n];

a(2^n) = 2 for all n >= 0.

A331745 Lexicographically earliest infinite sequence such that a(i) = a(j) => A278222(i) = A278222(j) and A323901(i) = A323901(j) for all i, j.

Original entry on oeis.org

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

Offset: 0

Antti Karttunen, Feb 04 2020

nonn

Restricted growth sequence transform of the ordered pair [A278222(n), A323901(n)].

Cf. A278222, A323901.

Cf. also A324400, A286622, A318310, A324389, A331744, A331746.

\\ Needs also code from A323901.
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; };
A005940(n) = { my(p=2, t=1); n--; until(!n\=2, if((n%2), (t*=p), p=nextprime(p+1)));
t };
A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); };  \\ From A046523
A278222(n) = A046523(A005940(1+n));
Aux331745(n) = [A278222(n),A323901(n)];
v331745 = rgs_transform(vector(1+up_to, n, Aux331745(n-1)));
A331745(n) = v331745[1+n];

a(2^n) = 2 for all n >= 0.

cp's OEIS Frontend

A336156 Lexicographically earliest infinite sequence such that a(i) = a(j) => A007814(1+i) = A007814(1+j) and A336158(i) = A336158(j), for all i, j >= 1.

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A351461 Lexicographically earliest infinite sequence such that a(i) = a(j) => A206787(i) = A206787(j) and A336651(i) = A336651(j) for all i, j >= 1.

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A336146 Lexicographically earliest infinite sequence such that a(i) = a(j) => A000035(i) = A000035(j) and A000265(i) = A000265(j), for all i, j >= 1.

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A336149 Lexicographically earliest infinite sequence such that a(i) = a(j) => A278221(i) = A278221(j) and A278222(i) = A278222(j), for all i, j >= 1.

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A336160 Lexicographically earliest infinite sequence such that a(i) = a(j) => A335915(i) = A335915(j) and A336158(i) = A336158(j), for all i, j >= 1.

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A336460 Lexicographically earliest infinite sequence such that a(i) = a(j) => f(i) = f(j), where f(n) = [A278222(n), A336158(n), A336466(n)], for all i, j >= 1.

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A351460 Lexicographically earliest infinite sequence such that a(i) = a(j) => A006530(i) = A006530(j), A206787(i) = A206787(j) and A336651(i) = A336651(j) for all i, j >= 1.

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A324530 Lexicographically earliest sequence such that a(i) = a(j) => f(i) = f(j), where f(n) = [A033879(n), A318458(n)] for all other numbers, except f(1) = -1.

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A331743 Lexicographically earliest infinite sequence such that a(i) = a(j) => A002487(i) = A002487(j) and A323901(i) = A323901(j) for all i, j.

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