A326075 -id:A326075 - cp's OEIS frontend

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

Antti Karttunen, Aug 23 2019

nonn

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.

			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

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

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

Cf. also A326196.

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([])));

a(p) = 1 for all primes p.

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

A326139 a(n) = gcd(A032742(n), A302042(n)).

Original entry on oeis.org

1, 1, 1, 2, 1, 3, 1, 4, 3, 5, 1, 6, 1, 7, 5, 8, 1, 9, 1, 10, 1, 11, 1, 12, 5, 13, 1, 14, 1, 15, 1, 16, 1, 17, 7, 18, 1, 19, 1, 20, 1, 21, 1, 22, 3, 23, 1, 24, 7, 25, 1, 26, 1, 27, 1, 28, 1, 29, 1, 30, 1, 31, 1, 32, 1, 33, 1, 34, 1, 35, 1, 36, 1, 37, 1, 38, 11, 39, 1, 40, 3, 41, 1, 42, 1, 43, 1, 44, 1, 45, 1, 46, 1, 47, 1, 48, 1, 49, 1, 50, 1, 51, 1, 52, 1

Offset: 1

Antti Karttunen, Aug 23 2019

nonn

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

Cf. A032742, A302042, A323888, A326075, A326189, A326190, A326191.

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));
A326139(n) = gcd(A032742(n),A302042(n));

a(n) = gcd(A032742(n), A302042(n)).

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).

Original entry on oeis.org

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

Antti Karttunen, Aug 23 2019

nonn

			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

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

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

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];

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).

A326191 Length of the longest 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).

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Aug 23 2019

nonn

			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 longest path(s) from 153 to 1 is 4 (there are actually two longest paths: 153 -> 51 -> 25 -> 5 -> 1 and 153 -> 75 -> 25 -> 5 -> 1), thus a(153) = 4.
.
        153
       /  \
      /    \
     51    75
     / \  /  \
    /   17    \
    \    |    /
     \   1   /
      \     /
       \   /
        25
         |
         5
         |
         1

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

Cf. A001222, A032742, A253557, A302042, A323888, A326075, A326139, A326189, A326190.

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));
A326191(n) = if(1==n,0,1+max(A326191(A032742(n)), A326191(A302042(n))));
\\ Slightly faster:
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)));
A326191list(up_to) = { my(v=vector(up_to)); v[1] = 0; for(n=2,up_to, v[n] = 1+max(v[A032742(n)], v[A302042(n)])); (v); };
v326191 = A326191list(up_to);
A326191(n) = v326191[n];

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

A326189(n) >= a(n) >= max(A001222(n), A253557(n)) >= min(A001222(n), A253557(n)) >= A326190(n).

cp's OEIS Frontend

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

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A326139 a(n) = gcd(A032742(n), A302042(n)).

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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).

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A326191 Length of the longest 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).

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