A300242 -id:A300242 - cp's OEIS frontend

A300240 Filter sequence combining A009195(n) and A046523(n), i.e., gcd(n,phi(n)) and the prime signature of n.

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

Offset: 1

Antti Karttunen, Mar 02 2018

nonn

Restricted growth sequence transform of P(A009195(n), A046523(n)), where P(a,b) is a two-argument form of A000027 used as a Cantor pairing function N x N -> N.

			a(6) = a(10) (= 4) because both 6 and 10 are nonsquare semiprimes, and A009195(6) = A009195(10) = 2.

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

Cf. A009195, A046523.

Cf. also A300230, A300241, A300242, 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])); }
A009195(n) = gcd(n, eulerphi(n));
A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); };  \\ From A046523
Aux300240(n) = (1/2)*(2 + ((A046523(n)+A009195(n))^2) - A046523(n) - 3*A009195(n));
write_to_bfile(1,rgs_transform(vector(65537,n,Aux300240(n))),"b300240.txt");

A300243 Filter sequence combining A051953(n) and A009195(n), n-phi(n) (cototient of n) and gcd(n,n-phi(n)).

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, 40, 41, 32, 42, 2, 43, 2, 44, 45, 46, 47, 48, 2, 49, 50, 48, 2, 51, 2, 52, 53, 54, 47, 55, 2, 56, 57, 58, 2, 59, 60, 61, 62, 63, 2, 64, 37, 65, 66, 67, 68, 69, 2, 70, 71

Offset: 1

Antti Karttunen, Mar 02 2018

nonn

Restricted growth sequence transform of P(A051953(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(66) = a(70) (= 48) because A051953(66) = A051953(70) = 46 and A009195(66) = A009195(70) = 2.

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

Cf. A009195, A051953.

Cf. also A300233, A300240, A300241, A300242.

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])); }
A009195(n) = gcd(n, eulerphi(n));
A051953(n) = (n - eulerphi(n));
Aux300243(n) = (1/2)*(2 + ((A051953(n)+A009195(n))^2) - A051953(n) - 3*A009195(n));
write_to_bfile(1,rgs_transform(vector(65537,n,Aux300243(n))),"b300243.txt");

A300241 Filter sequence combining A001065(n) and A009195(n), the sum of proper divisors of n and gcd(n,phi(n)).

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, 40, 41, 42, 43, 2, 44, 2, 45, 46, 47, 48, 49, 2, 50, 51, 52, 2, 53, 2, 54, 55, 56, 48, 57, 2, 58, 59, 60, 2, 61, 62, 63, 64, 65, 2, 66, 37, 67, 68, 69, 70, 71, 2, 72, 73

Offset: 1

Antti Karttunen, Mar 02 2018

nonn

Restricted growth sequence transform of P(A001065(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(65) = a(77) (= 48) because A001065(65) = A001065(77) = 19 and A009195(65) = A009195(77) = 1.

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

Cf. A001065, A009195.

Cf. also A300231, A300240, A300242, 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])); }
A001065(n) = (sigma(n)-n);
A009195(n) = gcd(n, eulerphi(n));
Aux300241(n) = (1/2)*(2 + ((A001065(n)+A009195(n))^2) - A001065(n) - 3*A009195(n));
write_to_bfile(1,rgs_transform(vector(65537,n,Aux300241(n))),"b300241.txt");

A326192 Lexicographically earliest sequence such that for all i, j, a(i) = a(j) => A009195(i) = A009195(j) and f(i) = f(j), where f(n) = gcd(n,sigma(n)) * (-1)^[gcd(n,sigma(n))==n] and A009195(n) = gcd(n, phi(n)).

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Aug 24 2019

nonn

Restricted growth sequence transform of the ordered pair [A009195(n), A326193(n)].
For all i, j:
  A305800(i) = A305800(j) => a(i) = a(j),
  a(i) = a(j) => A300242(i) = A300242(j),
  a(i) = a(j) => A326196(i) = A326196(j).

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

Cf. A009194, A009195, A300242, A305800, A326193, A326194, A326196.

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; };
Aux326192(n) = { my(u=gcd(n,sigma(n))); [gcd(n,eulerphi(n)), u*((-1)^(u==n))]; };
v326192 = rgs_transform(vector(up_to, n, Aux326192(n)));
A326192(n) = v326192[n];

cp's OEIS Frontend

A300240 Filter sequence combining A009195(n) and A046523(n), i.e., gcd(n,phi(n)) and the prime signature of n.

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A300243 Filter sequence combining A051953(n) and A009195(n), n-phi(n) (cototient of n) and gcd(n,n-phi(n)).

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A300241 Filter sequence combining A001065(n) and A009195(n), the sum of proper divisors of n and gcd(n,phi(n)).

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A326192 Lexicographically earliest sequence such that for all i, j, a(i) = a(j) => A009195(i) = A009195(j) and f(i) = f(j), where f(n) = gcd(n,sigma(n)) * (-1)^[gcd(n,sigma(n))==n] and A009195(n) = gcd(n, phi(n)).

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