A300243 -id:A300243 - 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");

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");

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");

A326198 Number of positive integers that are reachable from n with some combination of transitions x -> x-phi(x) and x -> gcd(x,phi(x)).

Original entry on oeis.org

1, 2, 2, 3, 2, 4, 2, 4, 3, 5, 2, 5, 2, 5, 3, 5, 2, 7, 2, 6, 4, 6, 2, 6, 3, 6, 4, 6, 2, 7, 2, 6, 3, 8, 3, 8, 2, 7, 5, 7, 2, 9, 2, 7, 5, 7, 2, 7, 3, 10, 3, 7, 2, 11, 5, 7, 5, 8, 2, 8, 2, 7, 5, 7, 3, 8, 2, 9, 4, 8, 2, 9, 2, 8, 5, 8, 3, 12, 2, 8, 5, 10, 2, 10, 5, 8, 3, 8, 2, 10, 3, 8, 5, 8, 3, 8, 2, 9, 6, 11, 2, 9, 2, 8, 6

Offset: 1

Antti Karttunen, Aug 24 2019

nonn

			From n = 12 we can reach any of the following numbers > 0: 12 (with an empty sequence of transitions), 8 (as A051953(12) = 8), 4 (as A009195(12) = A009195(8) = A051953(8) = 4), 2 (as A009195(4) = A051953(4) = 2) and 1 (as A009195(2) = A051953(2) = 1), thus a(12) = 5.
The directed acyclic graph formed from those two transitions with 12 as its unique root looks like this:
    12
    / \
   |   8
    \ /
     4
     |
     2
     |
     1

Antti Karttunen, Table of n, a(n) for n = 1..16384
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..100000

Cf. A000010, A009195, A051953, A071575, A300243, A326195.

Cf. also A326189, A326196, A327160, A327161.

A326198aux(n,xs) = if(vecsearch(xs,n),xs, xs = setunion([n],xs); if(1==n,xs, my(a=gcd(n,eulerphi(n)), b=n-eulerphi(n)); xs = A326198aux(a,xs); if((a==b),xs, A326198aux(b,xs))));
A326198(n) = length(A326198aux(n,Set([])));

a(n) > max(A071575(n), A326195(n)).

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

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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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A326198 Number of positive integers that are reachable from n with some combination of transitions x -> x-phi(x) and x -> gcd(x,phi(x)).

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