A326198 -id:A326198 - cp's OEIS frontend

A326196 Number of divisors of n that are reachable from n with some combination of transitions x -> gcd(x,sigma(x)) and x -> gcd(x,phi(x)).

1, 2, 2, 3, 2, 3, 2, 4, 3, 3, 2, 4, 2, 3, 3, 5, 2, 5, 2, 4, 3, 3, 2, 6, 3, 3, 4, 4, 2, 4, 2, 6, 3, 3, 2, 5, 2, 3, 3, 6, 2, 4, 2, 4, 3, 3, 2, 6, 3, 4, 3, 4, 2, 6, 3, 5, 3, 3, 2, 5, 2, 3, 4, 7, 2, 4, 2, 4, 3, 3, 2, 8, 2, 3, 3, 4, 2, 4, 2, 6, 5, 3, 2, 6, 2, 3, 3, 5, 2, 6, 3, 4, 3, 3, 3, 8, 2, 4, 3, 5, 2, 4, 2, 5, 3

Offset: 1

Antti Karttunen, Aug 24 2019

nonn

Number of distinct vertices in the directed acyclic graph formed by edge relations x -> A009194(x) and x -> A009195(x), where n is the unique root of the graph.
Because both A009194(n) and A009195(n) are divisors of n, it means that any number reached from n must also be a divisor of n. Number n is also included in the count as it is reached with an empty sequence of transitions.
Question: Are there any other numbers than those in A000961 for which a(n) = A000005(n) ?

			From n = 12 we can reach any of the following of its 6 divisors: 12 (with an empty combination of transitions), 4 (as A009194(12) = A009195(12) = 4), 2 (as A009195(4) = 2) and 1 (as A009194(4) = 1 = A009194(2) = A009195(2)), thus a(12) = 4.
The directed acyclic graph formed from those two transitions with 12 as its unique root looks like this:
   12
    |
    4
    | \
    |  2
    | /
    1

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

Cf. A000005, A000961, A009194, A009195, A326192, A326194, A326195, A326197.

Cf. also A326189, A326198.

A326196aux(n,xs) = { xs = setunion([n],xs); if(1==n,xs, my(a=gcd(n,eulerphi(n)), b=gcd(n,sigma(n))); xs = A326196aux(a,xs); if((a==b)||(b==n),xs, A326196aux(b,xs))); };
A326196(n) = length(A326196aux(n,Set([])));

a(n) = A000005(n) - A326197(n).

a(n) > max(A326194(n), A326195(n)).

A327160 Number of positive integers that are reachable from n with some combination of transitions x -> usigma(x)-x and x -> gcd(x,usigma(x)), where usigma is the sum of unitary divisors of n (A034448).

Original entry on oeis.org

1, 2, 2, 2, 2, 1, 2, 2, 2, 4, 2, 4, 2, 5, 4, 2, 2, 6, 2, 5, 3, 6, 2, 5, 2, 4, 2, 5, 2, 4, 2, 2, 5, 6, 3, 6, 2, 7, 3, 6, 2, 4, 2, 4, 5, 5, 2, 7, 2, 7, 5, 8, 2, 4, 3, 4, 3, 4, 2, 1, 2, 7, 3, 2, 3, 4, 2, 7, 4, 8, 2, 7, 2, 7, 3, 6, 3, 3, 2, 7, 2, 6, 2, 7, 3, 6, 6, 7, 2, 1, 5, 6, 4, 8, 4, 9, 2, 9, 5, 9, 2, 4, 2, 7, 7

Offset: 1

Antti Karttunen, Aug 25 2019

nonn

Question: Is this sequence well defined for every n ? If A318882 is not well defined in whole N, then neither this can be.

			From n = 30 we can reach any of the following strictly positive numbers: 30 (e.g., with an empty sequence of transitions), 42 (as A034460(30) = 42), 54 (as A034460(42) = 54; note that A034460(54) = 30 again) and 6 as A323166(30) = A323166(42) = A323166(54) = 6 = A323166(6) = A034460(6), thus a(30) = 4.

Antti Karttunen, Table of n, a(n) for n = 1..20000
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..100000

Cf. A034448, A034460, A318882, A323166.

Cf. also A326198, A327161.

A034448(n) = { my(f=factorint(n)); prod(k=1, #f~, 1+(f[k, 1]^f[k, 2])); }; \\ After code in A034448
A327160aux(n,xs) = if(vecsearch(xs,n),xs, xs = setunion([n],xs); if(1==n,xs, my(a=A034448(n)-n, b=gcd(A034448(n),n)); xs = A327160aux(a,xs); if((a==b),xs, A327160aux(b,xs))));
A327160(n) = length(A327160aux(n,Set([])));

A327161 Number of positive integers that are reachable from n with some combination of transitions x -> usigma(x)-x and x -> gcd(x,phi(x)), where usigma is the sum of unitary divisors of n (A034448), and phi is Euler totient function (A000010).

Original entry on oeis.org

1, 2, 2, 3, 2, 3, 2, 4, 3, 5, 2, 5, 2, 6, 4, 5, 2, 7, 2, 6, 4, 7, 2, 6, 3, 6, 4, 6, 2, 10, 2, 6, 5, 7, 3, 8, 2, 8, 4, 7, 2, 10, 2, 6, 5, 7, 2, 8, 3, 8, 5, 8, 2, 10, 4, 6, 4, 7, 2, 4, 2, 8, 5, 7, 3, 6, 2, 8, 5, 9, 2, 9, 2, 8, 4, 7, 3, 5, 2, 9, 5, 7, 2, 8, 3, 8, 6, 7, 2, 4, 5, 7, 5, 9, 4, 11, 2, 11, 5, 13, 2, 10, 2, 8, 7

Offset: 1

Antti Karttunen, Aug 25 2019

nonn

Question: Is this sequence well-defined for every n > 0? If A318882 is not well-defined for all positive integers, then neither can this be.

			a(30) = 10 as the graph obtained from vertex-relations x -> A034460(x) and x -> A009195(x) spans the following ten numbers [1, 2, 4, 6, 8, 12, 18, 30, 42, 54], which is illustrated below:
.
  30 -> 42 -> 54 (-> 30 ...)
   |     |     |
   2 <-- 6 <- 18
   |  \        |
   1 <-- 4 <- 12
            \  |
             <-8

Antti Karttunen, Table of n, a(n) for n = 1..20000
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..100000

Cf. A000010, A009195, A034448, A034460, A318882, A326195.

Cf. also A326196, A326198, A327160.

A034460(n) = (sumdivmult(n, d, if(gcd(d, n/d)==1, d))-n); \\ From A034460
A327161aux(n,xs) = if(vecsearch(xs,n),xs, xs = setunion([n],xs); if(1==n,xs, my(a=A034460(n), b=gcd(eulerphi(n),n)); xs = A327161aux(a,xs); if((a==b),xs, A327161aux(b,xs))));
A327161(n) = length(A327161aux(n,Set([])));

a(n) >= max(A318882(n), 1+A326195(n)).

cp's OEIS Frontend

A326196 Number of divisors of n that are reachable from n with some combination of transitions x -> gcd(x,sigma(x)) and x -> gcd(x,phi(x)).

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A327160 Number of positive integers that are reachable from n with some combination of transitions x -> usigma(x)-x and x -> gcd(x,usigma(x)), where usigma is the sum of unitary divisors of n (A034448).

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A327161 Number of positive integers that are reachable from n with some combination of transitions x -> usigma(x)-x and x -> gcd(x,phi(x)), where usigma is the sum of unitary divisors of n (A034448), and phi is Euler totient function (A000010).

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