A326189 Number of distinct nonnegative integers that are reachable from n with some nonempty combination of transitions x -> A032742(x) and x -> A302042(x).

0, 1, 1, 2, 1, 2, 1, 3, 2, 2, 1, 3, 1, 2, 2, 4, 1, 3, 1, 3, 4, 2, 1, 4, 2, 2, 4, 3, 1, 3, 1, 5, 4, 2, 2, 4, 1, 2, 3, 4, 1, 5, 1, 3, 7, 2, 1, 5, 2, 3, 4, 3, 1, 5, 4, 4, 6, 2, 1, 4, 1, 2, 6, 6, 3, 5, 1, 3, 6, 3, 1, 5, 1, 2, 4, 3, 2, 4, 1, 5, 8, 2, 1, 6, 4, 2, 4, 4, 1, 8, 4, 3, 9, 2, 3, 6, 1, 3, 6, 4, 1, 5, 1, 4, 7

Offset: 1

Author

Antti Karttunen, Aug 23 2019

Keywords

• nonn

Comments

Number of distinct numbers > 1 in the directed acyclic graph formed by edge relations x -> A032742(x) and x -> A302042(x), where n is the unique root of the graph.

Examples

			The directed acyclic graph whose unique root is 153 (illustrated below), spans the following seven numbers [1, 5, 17, 25, 51, 75, 153], as A032742(153) = 51, A302042(153) = 75, A032742(51) = 17, A302042(51) = 25, A032742(75) = 25, A302042(75) = 15, A032742(25) = A302042(25) = 5, and A032742(17) = A302042(17) = A032742(5) = A302042(5) = 1. We exclude the root 153 from the count of numbers that are reached, thus a(153) = 6. (Equally, we can include 153, but exclude 1).
.
        153
       /  \
      /    \
     51    75
     / \  /  \
    /   17    \
    \    |    /
     \   1   /
      \     /
       \   /
        25
         |
         5
         |
         1
		

Links

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

Crossrefs

Cf. A001222, A032742, A253557, A302042, A323888, A326075, A326139, A326190, A326191.

Cf. also A326196.

Programs

PARI

up_to = 65537;
ordinal_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), pt); for(i=1, length(invec), if(mapisdefined(om,invec[i]), pt = mapget(om, invec[i]), pt = 0); outvec[i] = (1+pt); mapput(om,invec[i],(1+pt))); outvec; };
A020639(n) = if(n>1, if(n>n=factor(n, 0)[1, 1], n, factor(n)[1, 1]), 1); \\ From A020639
A032742(n) = (n/A020639(n));
v078898 = ordinal_transform(vector(up_to,n,A020639(n)));
A078898(n) = v078898[n];
A302042(n) = if((1==n)||isprime(n),1,my(c = A078898(n), p = prime(-1+primepi(A020639(n))+primepi(A020639(c))), d = A078898(c), k=0); while(d, k++; if((1==k)||(A020639(k)>=p),d -= 1)); (k*p));
A326189aux(n,distvals) = if(1==n,distvals,my(newdistvals=setunion([n],distvals),a=A032742(n), b=A302042(n)); newdistvals = A326189aux(a,newdistvals); if(a==b,newdistvals, A326189aux(b,newdistvals)));
A326189(n) = length(A326189aux(n,Set([])));

Formula

a(p) = 1 for all primes p.

a(n) >= A326191(n) >= max(A001222(n),A253557(n)) >= min(A001222(n),A253557(n)) >= A326190(n).