A327160 -id:A327160

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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)).

+10
4

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

LINKS

Antti Karttunen, Table of n, a(n) for n = 1..16384

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

FORMULA

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

EXAMPLE

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:

/ \

| 8

\ /

PROG

(PARI)

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([])));

CROSSREFS

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

Cf. also A326189, A326196, A327160, A327161.

KEYWORD

nonn

AUTHOR

Antti Karttunen, Aug 24 2019

STATUS

approved

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).

+10
3

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

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.

LINKS

Antti Karttunen, Table of n, a(n) for n = 1..20000

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

FORMULA

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

EXAMPLE

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

PROG

(PARI)

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([])));

CROSSREFS

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

Cf. also A326196, A326198, A327160.