A369051 -id:A369051 - cp's OEIS frontend

A344025 Lexicographically earliest infinite sequence such that a(i) = a(j) => A003415(i) = A003415(j) and A003557(i) = A003557(j), for all i, j >= 1.

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, 28, 40, 41, 21, 2, 42, 2, 43, 44, 45, 46, 47, 2, 48, 49, 50, 2, 51, 2, 52, 53, 54, 46, 55, 2, 56, 57, 58, 2, 59, 41, 60, 61, 62, 2, 63, 37, 64

Offset: 1

Antti Karttunen, May 07 2021

nonn

Restricted growth sequence transform of the ordered pair [A003415(n), A003557(n)], where A003415(n) is the arithmetic derivative of n, and A003557(n) is n divided by its largest squarefree divisor.
For all i, j:
  parent(i) = parent(j) => a(i) = a(j),
  a(i) = a(j) => A342001(i) = A342001(j),
  a(i) = a(j) => A369051(i) = A369051(j) => A085731(i) = A085731(j).
Where "parent" can be any of the sequences A351236, A351260, A353520, A353521, A369050, for example.

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

Cf. A003415, A003557, A085731, A342001, A369051.

Cf. also A305800, A351236, A351260, A353520, A353521, A369050, and also A353523.

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) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A003557(n) = (n/factorback(factorint(n)[, 1]));
Aux344025(n) = [A003415(n), A003557(n)];
v344025 = rgs_transform(vector(up_to, n, Aux344025(n)));
A344025(n) = v344025[n];

A369050 Lexicographically earliest infinite sequence such that a(i) = a(j) => f(i) = f(j), where f(1) = 1, and for n>1, f(n) = [A003415(n), A369049(n)], for all i, j >= 1.

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, 28, 40, 41, 42, 2, 43, 2, 44, 45, 46, 47, 48, 2, 49, 50, 51, 2, 52, 2, 53, 54, 55, 56, 57, 2, 58, 59, 60, 2, 61, 62, 63, 64, 65, 2, 66, 37

Offset: 1

Antti Karttunen, Jan 15 2024

nonn

Restricted growth sequence transform of the function f defined as: f(1) = 1, and for n>1, f(n) = [A003415(n), A369049(n)].
For all i, j:
  A305800(i) = A305800(j) => a(i) = a(j),
  a(i) = a(j) => A344025(i) = A344025(j) => A369051(i) = A369051(j).

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

Cf. A003415, A369049.
Cf. also A305800, A328767, A344025, A369051.
Differs from A351260 for the first time at n=77, where a(77) = 56, while A351260(77) = 47.
Differs from A300833 for the first time at n=91, where a(91) = 37, while A300833(91) = 67.

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) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A369049(n) = (n % A003415(n));
Aux369050(n) = if(1==n,1,[A003415(n), A369049(n)]);
v369050 = rgs_transform(vector(up_to, n, Aux369050(n)));
A369050(n) = v369050[n];

A373150 Lexicographically earliest infinite sequence such that a(i) = a(j) => f(i) = f(j), where f(1) = 1, and for n>1, f(n) = [A003415(n), A085731(n), A373148(n)], for all i, j >= 1.

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, 28, 40, 41, 42, 2, 43, 2, 44, 45, 46, 47, 48, 2, 49, 50, 51, 2, 52, 2, 53, 54, 55, 56, 57, 2, 58, 59, 60, 2, 61, 62, 63, 64, 65, 2, 66, 67, 68, 69, 70, 71, 72, 2, 73, 74

Offset: 1

Antti Karttunen, May 27 2024

nonn

Restricted growth sequence transform of the function f defined as: f(1) = 1, and for n>1, f(n) = [A003415(n), A085731(n), A373148(n)].
For all i, j >= 1:
  A305800(i) = A305800(j) => a(i) = a(j),
  a(i) = a(j) => A369051(i) = A369051(j),
  a(i) = a(j) => A373151(i) = A373151(j) => A373143(i) = A373143(j).

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

Cf. A003415, A002110, A085731, A276085, A373148.
Cf. A305800, A369051, A373143, A373151.
Differs from A369050 for the first time at n=91, where a(91)=67, while A369050(91)=37.
Differs from A300833 for the first time at n=133, where a(133)=133, while A300833(133)=50.

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; };
A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A085731(n) = gcd(A003415(n),n);
A002110(n) = prod(i=1,n,prime(i));
A276085(n) = { my(f = factor(n)); sum(k=1, #f~, f[k, 2]*A002110(primepi(f[k, 1])-1)); };
A373148(n) = (A276085(n)%A003415(n));
Aux373150(n) = if(1==n,1,[A003415(n), A085731(n), A373148(n)]);
v373150 = rgs_transform(vector(up_to, n, Aux373150(n)));
A373150(n) = v373150[n];

A369046 Lexicographically earliest infinite sequence such that a(i) = a(j) => A003415(i) = A003415(j) and A069359(i) = A069359(j), for all i, j >= 1.

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, 28, 40, 41, 21, 2, 42, 2, 43, 44, 45, 46, 47, 2, 48, 49, 50, 2, 51, 2, 52, 53, 54, 46, 55, 2, 56, 57, 58, 2, 59, 41, 60, 61, 62, 2, 63, 37

Offset: 1

Antti Karttunen, Jan 18 2024

nonn

Restricted growth sequence transform of the ordered pair [A003415(n), A069359(n)].
For all i,j >= 1:
  A369050(i) = A369050(j) => a(i) = a(j), [Conjectured]
  a(i) = a(j) => A329039(i) = A329039(j) => A008966(i) = A008966(j).

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

Cf. A003415, A008966, A069359, A329039.
Cf. also A369050, A369051.
Differs from A351236 for the first time at n=91, where a(91) = 37, while A351236(91) = 64.
Differs from A344025 for the first time at n=140, where a(140) = 97, while A344025(140) = 92.

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) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A069359(n) = (n*sumdiv(n, d, isprime(d)/d));
Aux369046(n) = [A003415(n), A069359(n)];
v369046 = rgs_transform(vector(up_to, n, Aux369046(n)));
A369046(n) = v369046[n];

A369047 Lexicographically earliest infinite sequence such that a(i) = a(j) => A003415(i) = A003415(j) and A345000(i) = A345000(j), for all i, j >= 1.

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, 14, 17, 18, 19, 2, 20, 2, 21, 22, 23, 24, 25, 2, 26, 27, 28, 2, 29, 2, 30, 31, 32, 2, 33, 22, 34, 35, 36, 2, 37, 27, 38, 39, 20, 2, 38, 2, 40, 41, 42, 43, 44, 2, 45, 46, 47, 2, 48, 2, 31, 49, 21, 43, 50, 2, 51, 52, 53, 2, 54, 39, 34, 55, 56, 2, 57

Offset: 1

Antti Karttunen, Jan 21 2024

nonn

Restricted growth sequence transform of the ordered pair [A003415(n), A345000(n)], or equally, of the pair [A003415(n), A369038(n)], or equally, of the pair [A345000(n), A369038(n)].
For all i, j >= 1:
  A351236(i) = A351236(j) => a(i) = a(j),
  a(i) = a(j) => A369034(i) = A369034(j),
  a(i) = a(j) => A369036(i) = A369036(j).

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

Cf. A003415, A345000, A369034, A369036, A369038.

Cf. also A351236, A369051.

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) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
A345000(n) = gcd(A003415(n),A003415(A276086(n)));
Aux369047(n) = [A003415(n), A345000(n)];
v369047 = rgs_transform(vector(up_to, n, Aux369047(n)));
A369047(n) = v369047[n];

A369261 Lexicographically earliest infinite sequence such that a(i) = a(j) => A324644(i) = A324644(j) and A369445(i) = A369445(j), for all i, j >= 1.

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, 21, 22, 15, 23, 16, 20, 24, 25, 26, 27, 21, 28, 29, 30, 29, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 29, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 35, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 62, 53, 51, 70, 71, 72, 58, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 63

Offset: 1

Antti Karttunen, Jan 25 2024

nonn

Restricted growth sequence transform of the ordered pair [A324644(n), A369445(n)], or equally, of the pair [A000203(n), A324644(n)], or equally, of the pair [A000203(n), A369445(n)].

For all i, j >= 1: a(i) = a(j) => A286603(i) = A286603(j).

Cf. A000203, A276086, A324644, A369445.

Cf. also A286603, A291751, A369260 and A353523, A369047, A369051, A369062.

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; };
A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
A324644(n) = gcd(sigma(n),A276086(n));
Aux369261(n) = { my(u=A324644(n)); [u, sigma(n)/u]; };
v369261 = rgs_transform(vector(up_to, n, Aux369261(n)));
A369261(n) = v369261[n];

A373379 Lexicographically earliest infinite sequence such that a(i) = a(j) => A003415(i) = A003415(j), A085731(i) = A085731(j) and A107463(i) = A107463(j), for all i, j >= 1.

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, 28, 40, 41, 42, 2, 43, 2, 44, 45, 46, 47, 48, 2, 49, 50, 51, 2, 52, 2, 53, 54, 55, 47, 56, 2, 57, 58, 59, 2, 60, 41, 61, 62, 63, 2, 64, 37, 65, 66, 67, 68, 69, 2, 70, 71, 72

Offset: 1

Antti Karttunen, Jun 03 2024

nonn

Restricted growth sequence transform of the triple [A003415(n), A085731(n), A107463(n)].
For all i, j >= 1:
  A305800(i) = A305800(j) => a(i) = a(j),
  a(i) = a(j) => A369051(i) = A369051(j),
  a(i) = a(j) => A373363(i) = A373363(j),
  a(i) = a(j) => A373364(i) = A373364(j).
Starts to differ from A300235 at n=153. - R. J. Mathar, Jun 06 2024

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

Cf. A001414, A003415, A085731, A107463, A305800, A369051, A373363, A373364.
Differs from A305895, A327931, and A353560 for the first time at n=1610, where a(1610) = 1112, while A305895(1610) = A327931(1610) = A353560(1610) = 1210.
Cf. also A373150, A373152, A373380.

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; };
A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A085731(n) = gcd(A003415(n),n);
A001414(n) = ((n=factor(n))[, 1]~*n[, 2]);
A107463(n) = if(n<=1,n,if(isprime(n),1,A001414(n)));
Aux373379(n) = [A003415(n), A085731(n), A107463(n)];
v373379 = rgs_transform(vector(up_to, n, Aux373379(n)));
A373379(n) = v373379[n];

A373980 Lexicographically earliest infinite sequence such that a(i) = a(j) => f(i) = f(j), where f(n) = [A003415(n), A085731(n), A181819(n), A373247(n)], for all i, j >= 1.

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, 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, 66, 67, 68, 69, 70, 3, 71, 72

Offset: 1

Antti Karttunen, Jun 24 2024

nonn

Restricted growth sequence transform of the quadruple [A003415(n), A085731(n), A181819(n), A373247(n)].
For all i, j >= 1:
  A305801(i) = A305801(j) => a(i) = a(j),
  a(i) = a(j) => A300249(i) = A300249(j) => A101296(i) = A101296(j),
  a(i) = a(j) => A369051(i) = A369051(j),
  a(i) = a(j) => A373250(i) = A373250(j).

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

Cf. A003415, A085731, A181819, A373247.
Cf. A101296, A300249, A305801, A369051, A373250.
Differs from A353520 and A361021 first at n=130, where a(130) = 82, while A353520(130) = A361021(130) = 96.

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; };
A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A181819(n) = factorback(apply(e->prime(e),(factor(n)[,2])));
Aux373980(n) = { my(d=A003415(n), s=A181819(n)); [d, s, gcd(n,d), n%s]; };
v373980 = rgs_transform(vector(up_to, n, Aux373980(n)));
A373980(n) = v373980[n];

A374040 Lexicographically earliest infinite sequence such that a(i) = a(j) => f(i) = f(j), where f(n) = [A003415(n), A085731(n), A007814(n), A007949(n)], for all i, j >= 1.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Jul 01 2024

nonn

Restricted growth sequence transform of the quadruple [A003415(n), A085731(n), A007814(n), A007949(n)].
For all i, j >= 1:
  A305900(i) = A305900(j) => a(i) = a(j),
  a(i) = a(j) => A322026(i) = A322026(j),
  a(i) = a(j) => A369051(i) = A369051(j) => A083345(i) = A083345(j),
  a(i) = a(j) => b(i) = b(j), where b can be any of the sequences listed at the crossrefs-section, under "some of the other matched sequences".

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

Cf. A003415, A007814, A007949, A085731.
Some of the other matched sequences (see comments): A083345, A359430, A369001, A369004, A369643, A369658, A373143, A373474, A373483.
Cf. also A322026, A353521, A369051, A373268, A372573, A374131 for similar and related constructions.
Differs from A305900 first at n=77, where a(77) = 50, while A305900(77) = 59.

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; };
A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
Aux374040(n) = { my(d=A003415(n)); [d, gcd(n,d), valuation(n,2), valuation(n,3)]; };
v374040 = rgs_transform(vector(up_to, n, Aux374040(n)));
A374040(n) = v374040[n];

A369040 Lexicographically earliest infinite sequence such that a(i) = a(j) => A003415(i) = A003415(j) and A342014(i) = A342014(j), for all i, j >= 1.

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, 14, 17, 18, 19, 2, 20, 2, 21, 22, 23, 24, 25, 2, 26, 27, 28, 2, 29, 2, 30, 31, 32, 2, 33, 22, 34, 35, 36, 2, 37, 27, 38, 39, 40, 2, 41, 2, 42, 43, 44, 45, 46, 2, 47, 48, 49, 2, 50, 2, 31, 51, 52, 45, 53, 2, 54, 55, 56, 2, 57, 39, 34, 58, 59, 2, 60, 35

Offset: 1

Antti Karttunen, Jan 18 2024

nonn

Restricted growth sequence transform of the ordered pair [A003415(n), A342014(n)].
For all i, j >= 1:
  A369050(i) = A369050(j) => a(i) = a(j), [Conjectured]
  a(i) = a(j) => A341625(i) = A341625(j).

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

Cf. A003415, A341625, A342014.

Cf. also A369040, A369050, A369051.

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; };
A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A342014(n) = (A003415(n)%n);
Aux369040(n) = [A003415(n), A342014(n)];
v369040 = rgs_transform(vector(up_to, n, Aux369040(n)));
A369040(n) = v369040[n];

cp's OEIS Frontend

A344025 Lexicographically earliest infinite sequence such that a(i) = a(j) => A003415(i) = A003415(j) and A003557(i) = A003557(j), for all i, j >= 1.

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A369050 Lexicographically earliest infinite sequence such that a(i) = a(j) => f(i) = f(j), where f(1) = 1, and for n>1, f(n) = [A003415(n), A369049(n)], for all i, j >= 1.

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A373150 Lexicographically earliest infinite sequence such that a(i) = a(j) => f(i) = f(j), where f(1) = 1, and for n>1, f(n) = [A003415(n), A085731(n), A373148(n)], for all i, j >= 1.

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A369046 Lexicographically earliest infinite sequence such that a(i) = a(j) => A003415(i) = A003415(j) and A069359(i) = A069359(j), for all i, j >= 1.

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A369047 Lexicographically earliest infinite sequence such that a(i) = a(j) => A003415(i) = A003415(j) and A345000(i) = A345000(j), for all i, j >= 1.

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A369261 Lexicographically earliest infinite sequence such that a(i) = a(j) => A324644(i) = A324644(j) and A369445(i) = A369445(j), for all i, j >= 1.

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A373379 Lexicographically earliest infinite sequence such that a(i) = a(j) => A003415(i) = A003415(j), A085731(i) = A085731(j) and A107463(i) = A107463(j), for all i, j >= 1.

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A373980 Lexicographically earliest infinite sequence such that a(i) = a(j) => f(i) = f(j), where f(n) = [A003415(n), A085731(n), A181819(n), A373247(n)], for all i, j >= 1.

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A374040 Lexicographically earliest infinite sequence such that a(i) = a(j) => f(i) = f(j), where f(n) = [A003415(n), A085731(n), A007814(n), A007949(n)], for all i, j >= 1.

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