A326190 Length of the shortest path to 1 when starting from x=n and on each iteration step one may always choose either transition x -> A032742(x) or x -> A302042(x).

0, 1, 1, 2, 1, 2, 1, 3, 2, 2, 1, 3, 1, 2, 2, 4, 1, 3, 1, 3, 2, 2, 1, 4, 2, 2, 2, 3, 1, 3, 1, 5, 2, 2, 2, 4, 1, 2, 2, 4, 1, 3, 1, 3, 3, 2, 1, 5, 2, 3, 2, 3, 1, 3, 2, 4, 2, 2, 1, 4, 1, 2, 2, 6, 2, 3, 1, 3, 2, 3, 1, 5, 1, 2, 2, 3, 2, 3, 1, 5, 3, 2, 1, 4, 2, 2, 2, 4, 1, 4, 2, 3, 2, 2, 2, 6, 1, 3, 2, 4, 1, 3, 1, 4, 3

Offset

1,4

Links

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

Formula

a(1) = 0; for n > 1, a(n) = 1 + min(a(A032742(n)), a(A302042(n))).

a(n) <= min(A001222(n),A253557(n)) <= max(A001222(n),A253557(n)) <= A326191(n) <= A326189(n).

Example

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. The length of shortest path(s) from 153 to 1 is 3 (there are actually two shortest paths: 153 -> 51 -> 17 -> 1 and 153 -> 75 -> 17 -> 1), thus a(153) = 3.
.
        153
       /  \
      /    \
     51    75
     / \  /  \
    /   17    \
    \    |    /
     \   1   /
      \     /
       \   /
        25
         |
         5
         |
         1

Prog

(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; };
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; };
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));
A326190(n) = if(1==n, 0, 1+min(A326190(A032742(n)), A326190(A302042(n))));
\\ Somewhat faster version:
memo302042 = Map();
A302042(n) = if((1==n)||isprime(n), 1, my(v); if(mapisdefined(memo302042, n, &v), v, 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)); v=(k*p); mapput(memo302042, n, v); (v)));
A326190list(up_to) = { my(v=vector(up_to)); v[1] = 0; for(n=2, up_to, v[n] = 1+min(v[A032742(n)], v[A302042(n)])); (v); };
v326190 = A326190list(up_to);
A326190(n) = v326190[n];

Crossrefs

Cf. A032742, A253557, A302042, A323888, A326075, A326139, A326089, A326191.

Sequence in context: A179953 A277013 A305822 * A086436 A001222 A257091

Adjacent sequences: A326187 A326188 A326189 * A326191 A326192 A326193

Keyword

nonn

Author

Antti Karttunen, Aug 23 2019

Status

approved