A369050 -id:A369050 - 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];

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.

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, 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];

A369049 a(n) = n mod n', where n' is the arithmetic derivative of n, A003415.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 8, 3, 3, 0, 12, 0, 5, 7, 16, 0, 18, 0, 20, 1, 9, 0, 24, 5, 11, 0, 28, 0, 30, 0, 32, 5, 15, 11, 36, 0, 17, 7, 40, 0, 1, 0, 44, 6, 21, 0, 48, 7, 5, 11, 52, 0, 54, 7, 56, 13, 27, 0, 60, 0, 29, 12, 64, 11, 5, 0, 68, 17, 11, 0, 72, 0, 35, 20, 76, 5, 7, 0, 80, 81, 39, 0, 84, 19, 41, 23, 88, 0, 90, 11

Offset: 2

Antti Karttunen, Jan 15 2024

nonn

Antti Karttunen, Table of n, a(n) for n = 2..65537

Cf. A003415, A085731 [= gcd(A003415(n), a(n))].

Cf. also A328382, A342014, A369050.

Array[Mod[#, #  Total[#2/#1 & @@@ FactorInteger[#]]] &, 120, 2] (* Michael De Vlieger, Jan 15 2024 *)

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));

a(n) = n mod A003415(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];

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

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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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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A369049 a(n) = n mod n', where n' is the arithmetic derivative of n, A003415.

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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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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.

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