cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-7 of 7 results.

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)).

Original entry on oeis.org

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

Views

Author

Antti Karttunen, Aug 24 2019

Keywords

Comments

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) ?

Examples

			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
		

Crossrefs

Programs

  • PARI
    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([])));

Formula

a(n) = A000005(n) - A326197(n).
a(n) > max(A326194(n), A326195(n)).

A326075 Difference between the number of prime divisors in a nonstandard factorization process based on the sieve of Eratosthenes vs. their number in the ordinary factorization of n (when counted with multiplicity): a(n) = A253557(n) - A001222(n).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, -1, 1, 0, 1, 0, 0, 0, 0, 0, -1, 0, 0, 1, 0, 0, 2, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, -1, 0, 0, 1, 1, 0, 1, 0, 0, 1, 1, 0, 3, 0, 0, 0, 0, 0, -1, 0, 0, 1, 0, 0, 1
Offset: 1

Views

Author

Antti Karttunen, Aug 23 2019

Keywords

Examples

			A001222(21) = 2 because A032742(21) = 7, and A032742(7) = 1, while A253557(21) = 3 because A302042(21) = 9, A302042(9) = 3, and A302042(3) = 1. Thus a(21) = 3-2 = 1.
A001222(27) = 3 because A032742(27) = 9, A032742(9) = 3 and A032742(3) = 1, while A253557(27) = 2 because A302042(27) = 7 and A302042(7) = 1. Thus a(27) = 2-3 = -1.
		

Crossrefs

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
    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));
    A253557(n) = if(1==n, 0,1+A253557(A302042(n)));
    A326075(n) = (A253557(n)-bigomega(n));

Formula

a(n) = A253557(n) - A001222(n) = A001222(A250246(n)) - A001222(n).
a(p) = 0 for all primes p.

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

Views

Author

Antti Karttunen, Aug 23 2019

Keywords

Crossrefs

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; };
    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));

Formula

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

Views

Author

Antti Karttunen, Aug 23 2019

Keywords

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. 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
		

Crossrefs

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; };
    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];

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).

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

Views

Author

Antti Karttunen, Aug 23 2019

Keywords

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. 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
		

Crossrefs

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; };
    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];

Formula

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).

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)).

Original entry on oeis.org

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
Offset: 1

Views

Author

Antti Karttunen, Aug 24 2019

Keywords

Examples

			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:
    12
    / \
   |   8
    \ /
     4
     |
     2
     |
     1
		

Crossrefs

Programs

  • 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([])));

Formula

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

A326188 a(n) = A001065(n) - A003557(n), where A001065(n) = the sum of proper divisors of n, and A003557(n) = n divided by its largest squarefree divisor.

Original entry on oeis.org

-1, 0, 0, 1, 0, 5, 0, 3, 1, 7, 0, 14, 0, 9, 8, 7, 0, 18, 0, 20, 10, 13, 0, 32, 1, 15, 4, 26, 0, 41, 0, 15, 14, 19, 12, 49, 0, 21, 16, 46, 0, 53, 0, 38, 30, 25, 0, 68, 1, 38, 20, 44, 0, 57, 16, 60, 22, 31, 0, 106, 0, 33, 38, 31, 18, 77, 0, 56, 26, 73, 0, 111, 0, 39, 44, 62, 18, 89, 0, 98, 13, 43, 0, 138, 22, 45, 32, 88, 0, 141, 20
Offset: 1

Views

Author

Antti Karttunen, Jul 11 2019

Keywords

Crossrefs

Programs

  • PARI
    A003557(n) = { my(f=factor(n)); for (i=1, #f~, f[i, 2] = f[i, 2]-1); factorback(f); }; \\ From A003557
    A326188(n) = ((sigma(n)-A003557(n))-n);

Formula

a(n) = A326187(n) - n = A000203(n) - A003557(n) - n.
a(n) = A001065(n) - A003557(n).
Showing 1-7 of 7 results.