A369067 -id:A369067 - cp's OEIS frontend

A369065 Lexicographically earliest infinite sequence such that a(i) = a(j) => A344026(i) = A344026(j) for all i, j >= 0.

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

Offset: 0

Antti Karttunen, Jan 16 2024

nonn

Restricted growth sequence transform of A344026, i.e., of the arithmetic derivative (A003415) as reordered by the Doudna sequence (A005940).

Antti Karttunen, Table of n, a(n) for n = 0..65537

Cf. A003415, A005940, A344026.

Cf. also A366802, A369067.

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]));
A005940(n) = { my(p=2, t=1); n--; until(!n\=2, if((n%2), (t*=p), p=nextprime(p+1))); (t); };
A344026(n) = A003415(A005940(1+n));
v369065 = rgs_transform(vector(1+up_to,n,A344026(n-1)));
A369065(n) = v369065[1+n];

A369066 Dirichlet convolution of Liouville's lambda (A008836) with A083345, as reordered by the Doudna sequence.

Original entry on oeis.org

0, 1, 1, 0, 1, 3, 1, 3, 1, 5, 6, 0, 1, 2, 0, -1, 1, 7, 8, 0, 10, 14, 5, 8, 1, 2, 3, 0, 2, 0, 4, 6, 1, 11, 12, 0, 14, 20, 7, 14, 16, 34, 44, 0, 7, 9, 0, -3, 1, 2, 3, 0, 5, 5, 2, 5, 2, 6, 8, 0, 2, 7, 1, -3, 1, 13, 14, 0, 16, 32, 11, 20, 18, 54, 68, 0, 11, 13, 0, -5, 22, 76, 92, 0, 124, 92, 34, 36, 11, 13, 20, 0, 16

Offset: 0

Antti Karttunen, Jan 16 2024

Antti Karttunen, Table of n, a(n) for n = 0..16384

Cf. A005940, A008836, A083345, A369067 (rgs-transform), A369069.

Cf. also A366803 (compare the scatter plots).

A005940(n) = { my(p=2, t=1); n--; until(!n\=2, if((n%2), (t*=p), p=nextprime(p+1))); (t); };
A008836(n) = ((-1)^bigomega(n));
A083345(n) = { my(f=factor(n)); numerator(vecsum(vector(#f~, i, f[i, 2]/f[i, 1]))); };
A369069(n) = sumdiv(n,d,A008836(n/d)*A083345(d));
A369066(n) = A369069(A005940(1+n));

a(n) = A369069(A005940(1+n)).

A369457 Lexicographically earliest infinite sequence such that a(i) = a(j) => A369456(i) = A369456(j) for all i, j >= 0.

Original entry on oeis.org

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

Offset: 0

Antti Karttunen, Jan 27 2024

nonn

Restricted growth sequence transform of A369456.

Antti Karttunen, Table of n, a(n) for n = 0..65537

Cf. A005940, A083345, A369456.

Cf. also A366802, A369065, A369067.

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); };
A083345(n) = { my(f=factor(n)); numerator(vecsum(vector(#f~, i, f[i, 2]/f[i, 1]))); };
A369456(n) = A083345(A005940(1+n));
v369457 = rgs_transform(vector(1+up_to,n,A369456(n-1)));
A369457(n) = v369457[1+n];

cp's OEIS Frontend

A369065 Lexicographically earliest infinite sequence such that a(i) = a(j) => A344026(i) = A344026(j) for all i, j >= 0.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

A369066 Dirichlet convolution of Liouville's lambda (A008836) with A083345, as reordered by the Doudna sequence.

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

Formula

A369457 Lexicographically earliest infinite sequence such that a(i) = a(j) => A369456(i) = A369456(j) for all i, j >= 0.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI