A300225 -id:A300225 - cp's OEIS frontend

A300223 Lexicographically earliest infinite sequence such that a(i) = a(j) => A046523(i) = A046523(j) and A296091(i) = A296091(j), for all i, j >= 1.

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

Offset: 1

Antti Karttunen, Mar 01 2018

nonn

Restricted growth sequence transform of the ordered pair [A046523(n), A296091(n)].

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

Cf. A046523, A296091.

Cf. also A305800, A296090, A300224, A300225.

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; };
write_to_bfile(start_offset,vec,bfilename) = { for(n=1, length(vec), write(bfilename, (n+start_offset)-1, " ", vec[n])); }
A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); };  \\ From A046523
A296091(n) = if(1==n,n,A046523(sigma(n)-1));
Aux300223(n) = (1/2)*(2 + ((A046523(n)+A296091(n))^2) - A046523(n) - 3*A296091(n));
v300223 = rgs_transform(vector(up_to,n,Aux300223(n)));
A300223(n) = v300223[n];

Name changed by Antti Karttunen, May 20 2022

A300242 Filter sequence combining gcd(n,sigma(n)) and gcd(n,phi(n)), (A009194 and A009195).

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Mar 02 2018

nonn

Restricted growth sequence transform of P(A009194(n), A009195(n)), where P(a,b) is a two-argument form of A000027 used as a Cantor pairing function N x N -> N.

			a(15) = a(33) (= 8) because A009194(15) = A009194(33) = 3 and A009195(15) = A009195(33) = 1.
a(20) = a(52) (= 11) because A009194(20) = A009194(52) = 2 and A009195(20) = A009195(52) = 4.

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

Cf. A009194, A009195.

Cf. also A286591, A300225, A300240, A300241, A300243.

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; };
write_to_bfile(start_offset,vec,bfilename) = { for(n=1, length(vec), write(bfilename, (n+start_offset)-1, " ", vec[n])); }
A009194(n) = gcd(n, sigma(n));
A009195(n) = gcd(n, eulerphi(n));
Aux300242(n) = (1/2)*(2 + ((A009194(n)+A009195(n))^2) - A009194(n) - 3*A009195(n));
write_to_bfile(1,rgs_transform(vector(65537,n,Aux300242(n))),"b300242.txt");

A300224 Filter sequence combining A046523(n) and A296078(n), prime signature of n and prime signature of phi(n)+1.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Mar 01 2018

nonn

Restricted growth sequence transform of P(A046523(n), A296078(n)), where P(a,b) is a two-argument form of A000027 used as a Cantor pairing function N x N -> N.

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

Cf. A000010, A046523, A296078.

Cf. also A286160, A296085, A300223, A300225.

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; };
write_to_bfile(start_offset,vec,bfilename) = { for(n=1, length(vec), write(bfilename, (n+start_offset)-1, " ", vec[n])); }
A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); };  \\ This function from Charles R Greathouse IV, Aug 17 2011
A296078(n) = A046523(1+eulerphi(n));
Aux300224(n) = (1/2)*(2 + ((A296078(n)+A046523(n))^2) - A296078(n) - 3*A046523(n));
write_to_bfile(1,rgs_transform(vector(up_to,n,Aux300224(n))),"b300224.txt");

cp's OEIS Frontend

A300223 Lexicographically earliest infinite sequence such that a(i) = a(j) => A046523(i) = A046523(j) and A296091(i) = A296091(j), for all i, j >= 1.

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A300242 Filter sequence combining gcd(n,sigma(n)) and gcd(n,phi(n)), (A009194 and A009195).

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A300224 Filter sequence combining A046523(n) and A296078(n), prime signature of n and prime signature of phi(n)+1.

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