A085731 -id:A085731 - cp's OEIS frontend

A369051 Lexicographically earliest infinite sequence such that a(i) = a(j) => A003415(i) = A003415(j) and A085731(i) = A085731(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, 40, 2, 42, 43, 44, 45, 46, 2, 47, 48, 49, 2, 50, 2, 51, 52, 53, 45, 54, 2, 55, 56, 57, 2, 58, 41, 59, 60, 61, 2, 62, 37

Offset: 1

Antti Karttunen, Jan 15 2024

nonn

Restricted growth sequence transform of the ordered pair [A003415(n), A085731(n)], or equally, of the pair [A003415(n), A083345(n)], or equally, of the pair [A083345(n), A085731(n)].

For all i, j: A369050(i) = A369050(j) => A344025(i) = A344025(j) => a(i) = a(j).

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

Cf. A003415, A083345, A085731.

Cf. also A344025, A369050.

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]));
A085731(n) = gcd(A003415(n),n);
Aux369051(n) = [A003415(n), A085731(n)];
v369051 = rgs_transform(vector(up_to, n, Aux369051(n)));
A369051(n) = v369051[n];

A369008 a(n) = A085731(n) / A003557(n).

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 3, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 3, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1

Offset: 1

Antti Karttunen, Jan 15 2024

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

Cf. A003415, A003557, A083345, A085731, A342001, A369009.

Cf. A342090 (positions of terms > 1).

f[p_, e_] := If[Divisible[e, p], p, 1]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Jan 20 2024 *)

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]));
A369008(n) = { my(u=A003415(n)); (gcd(n,u)/A003557(n)); };

A369008(n) = if(1==n, n, my(f=factor(n)); for(i=1, #f~, if((f[i, 2]%f[i, 1]), f[i, 1] = 1, f[i, 2] = 1)); factorback(f));

Multiplicative with a(p^e) = p if p|e, otherwise a(p^e) = 1.
For n > 1, a(n) = A342001(n) / A083345(n).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Product_{p prime} ((p^(p+1) + p^2 - 3*p +1)/(p*(p^p-1))) = 1.22775972725472961826... . - Amiram Eldar, Jan 20 2024

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];

A359577 Dirichlet inverse of A085731, where A085731 is the greatest common divisor of n and the arithmetic derivative of n.

Original entry on oeis.org

1, -1, -1, -3, -1, 1, -1, 3, -2, 1, -1, 3, -1, 1, 1, -3, -1, 2, -1, 3, 1, 1, -1, -3, -4, 1, -22, 3, -1, -1, -1, 3, 1, 1, 1, 6, -1, 1, 1, -3, -1, -1, -1, 3, 2, 1, -1, 3, -6, 4, 1, 3, -1, 22, 1, -3, 1, 1, -1, -3, -1, 1, 2, -3, 1, -1, -1, 3, 1, -1, -1, -6, -1, 1, 4, 3, 1, -1, -1, 3, 28, 1, -1, -3, 1, 1, 1, -3, -1, -2, 1, 3, 1, 1, 1, -3, -1, 6, 2, 12, -1, -1, -1, -3, -1

Offset: 1

Antti Karttunen, Jan 06 2023

Multiplicative because A085731 is.

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

Cf. A003415, A085731, A038838 (positions of even terms), A122132 (of odd terms), A353627 (parity of terms).

g:= proc(n) option remember;
      igcd(n, n*add(i[2]/i[1], i=ifactors(n)[2]))
    end:
a:= proc(n) option remember; `if`(n=1, 1, -add(
      a(d)*g(n/d), d=numtheory[divisors](n) minus {n}))
    end:
seq(a(n), n=1..120);  # Alois P. Heinz, Jan 07 2023

d[0] = d[1] = 0; d[n_] := n * Plus @@ ((Last[#]/First[#]) & /@ FactorInteger[n]); s[n_] := GCD[n, d[n]]; a[1] = 1; a[n_] := a[n] = -DivisorSum[n, s[n/#]*a[#] &, # < n &]; Array[a, 100] (* Amiram Eldar, Jan 07 2023 *)

a(1) = 1, and for n > 1, a(n) = -Sum_{d|n, dA085731(n/d) * a(d).

A373152 Lexicographically earliest infinite sequence such that a(i) = a(j) => A085731(i) = A085731(j) and A373145(i) = A373145(j), for all i, j >= 1.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Jun 03 2024

nonn

Restricted growth sequence transform of the ordered pair [A085731(n), A373145(n)], i.e., the ordered pair [gcd(n, A003415(n)), gcd(A003415(n), A276085(n))].

For all i, j >= 1: A373150(i) = A373150(j) => a(i) = a(j).

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

Cf. A085731, A276085, A373145.

Cf. also A373150, A373151.

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]));
A276085(n) = { my(f = factor(n)); sum(k=1, #f~, f[k, 2]*prod(i=1,primepi(f[k, 1]-1),prime(i))); };
Aux373152(n) = [gcd(n, A003415(n)), gcd(A003415(n), A276085(n))];
v373152 = rgs_transform(vector(up_to, n, Aux373152(n)));
A373152(n) = v373152[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];

A268398 Partial sums of A085731.

Original entry on oeis.org

1, 2, 3, 7, 8, 9, 10, 14, 17, 18, 19, 23, 24, 25, 26, 42, 43, 46, 47, 51, 52, 53, 54, 58, 63, 64, 91, 95, 96, 97, 98, 114, 115, 116, 117, 129, 130, 131, 132, 136, 137, 138, 139, 143, 146, 147, 148, 164, 171, 176, 177, 181, 182, 209, 210, 214, 215, 216, 217

Offset: 1

Peter Kagey, Feb 03 2016

nonn

Peter Kagey, Table of n, a(n) for n = 1..10000
Project Euler, Problem 484: Arithmetic Derivative

Cf. A003415, A085731.

Accumulate@ Table[GCD[n, If[Abs@ n < 2, 0, n Total[#2/#1 & @@@ FactorInteger@ Abs@ n]]], {n, 58}] (* Michael De Vlieger, Feb 14 2016, after Michael Somos at A003415  *)
Accumulate@ Table[GCD[n, If[Abs@ n < 2, 0, n Total[#2/#1 & @@@ FactorInteger@ Abs@ n]]], {n, 58}] (* Michael De Vlieger, Feb 14 2016 *)

a085731(n) = {my(f = factor(n)); for (i=1, #f~, if (f[i,2] % f[i,1], f[i,2]--);); factorback(f);}
a(n) = sum(k=1, n, a085731(k)); \\ Michel Marcus, Feb 14 2016

require 'prime'
def a003415(n)
  return 0 if n == 1
  return 1 if Prime.prime?(n)
  a = Prime.each.find { |i| n % i == 0 }
  a * a003415(n/a) + n/a * a003415(a)
end
def a268398(n)
  sum = 0
  (1..n).map { |n| sum += a003415(n).gcd(n) }.last
end

A373268 Lexicographically earliest infinite sequence such that a(i) = a(j) => A003415(i) = A003415(j), A085731(i) = A085731(j) and A373145(i) = A373145(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, 64, 65, 66, 67, 68, 69, 2, 70, 71, 72, 2, 73, 2, 74, 55

Offset: 1

Antti Karttunen, Jun 09 2024

nonn

Restricted growth sequence transform of the triple [A003415(n), A085731(n), A373145(n)].
For all i, j >= 1:
  A373150(i) = A373150(j) => a(i) = a(j),
  a(i) = a(j) => A373151(i) = A373151(j) => A373485(i) = A373485(j),
  a(i) = a(j) => A373152(i) = A373152(j),
  a(i) = a(j) => A373486(i) = A373486(j).

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

Cf. A003415, A083345, A085731, A276085, A373145.
Cf. A373150, A373151, A373152, A373485, A373486.
Differs from A344025 and A369046 for the first time at n=91, where a(91) = 64, while A344025(91) = A369046(91) = 37.
Differs from A351236 for the first time at n=143, where a(143) = 100, while A351236(143) = 68.

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]));
A276085(n) = { my(f = factor(n)); sum(k=1, #f~, f[k, 2]*prod(i=1,primepi(f[k, 1]-1),prime(i))); };
Aux373268(n) = { my(d=A003415(n)); [d, gcd(d,n), gcd(d, A276085(n))]; };
v373268 = rgs_transform(vector(up_to, n, Aux373268(n)));
A373268(n) = v373268[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];

cp's OEIS Frontend

A369051 Lexicographically earliest infinite sequence such that a(i) = a(j) => A003415(i) = A003415(j) and A085731(i) = A085731(j), for all i, j >= 1.

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PARI

A369008 a(n) = A085731(n) / A003557(n).

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Mathematica

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PARI

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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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PARI

A359577 Dirichlet inverse of A085731, where A085731 is the greatest common divisor of n and the arithmetic derivative of n.

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Maple

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A373152 Lexicographically earliest infinite sequence such that a(i) = a(j) => A085731(i) = A085731(j) and A373145(i) = A373145(j), for all i, j >= 1.

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PARI

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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PARI

A268398 Partial sums of A085731.

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Mathematica

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Ruby

A373268 Lexicographically earliest infinite sequence such that a(i) = a(j) => A003415(i) = A003415(j), A085731(i) = A085731(j) and A373145(i) = A373145(j), for all i, j >= 1.

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PARI

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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