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.

A366874 Lexicographically earliest infinite sequence such that a(i) = a(j) => A366873(i) = A366873(j) for all i, j >= 0, where A366873 is the average of number of and sum of odd divisors of n as permuted by A163511.

Original entry on oeis.org

1, 1, 1, 2, 1, 3, 2, 4, 1, 5, 3, 6, 2, 7, 4, 8, 1, 9, 5, 10, 3, 11, 6, 12, 2, 13, 7, 14, 4, 15, 8, 16, 1, 17, 9, 18, 5, 19, 10, 20, 3, 21, 11, 22, 6, 23, 12, 24, 2, 25, 13, 26, 7, 27, 14, 28, 4, 29, 15, 30, 8, 14, 16, 3, 1, 31, 17, 32, 9, 33, 18, 34, 5, 35, 19, 36, 10, 37, 20, 38, 3, 39, 21, 40, 11, 41, 22, 42, 6
Offset: 0

Views

Author

Antti Karttunen, Oct 27 2023

Keywords

Comments

Restricted growth sequence transform of A366873.

Links

Crossrefs

Cf. A113415, A163511, A366873.
Cf. also A366798, A366806, A366878.

Programs

  • PARI
    up_to = 65537;
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; };
A113415(n) = if(n<1, 0, sumdiv(n, d, if(d%2, (d+1)/2)));
A163511(n) = if(!n, 1, my(p=2, t=1); while(n>1, if(!(n%2), (t*=p), p=nextprime(1+p)); n >>= 1); (t*p));
A366873(n) = A113415(A163511(n));
v366874 = rgs_transform(vector(1+up_to,n,A366873(n-1)));
A366874(n) = v366874[1+n];

A366881 Lexicographically earliest infinite sequence such that a(i) = a(j) => A206787(A163511(i)) = A206787(A163511(j)) and A336651(A163511(n)) = A336651(A163511(j)) for all i, j >= 0.

Original entry on oeis.org

1, 1, 1, 2, 1, 3, 2, 4, 1, 5, 3, 6, 2, 7, 4, 8, 1, 9, 5, 10, 3, 11, 6, 12, 2, 13, 7, 14, 4, 15, 8, 16, 1, 17, 9, 18, 5, 19, 10, 20, 3, 21, 11, 22, 6, 23, 12, 24, 2, 25, 13, 26, 7, 27, 14, 28, 4, 29, 15, 30, 8, 14, 16, 31, 1, 32, 17, 33, 9, 34, 18, 35, 5, 36, 19, 37, 10, 38, 20, 39, 3, 40, 21, 41, 11, 42, 22, 43, 6
Offset: 0

Views

Author

Antti Karttunen, Nov 04 2023

Keywords

Comments

Restricted growth sequence transform of the ordered pair [A206787(A163511(n)), A336651(A163511(n))].
Restricted growth sequence transform of sequence b(n) = A351461(A163511(n)).
For all i, j >= 0:
a(i) = a(j) => A324186(i) = A324186(j), (similarly for A366806)
a(i) = a(j) => A366885(i) = A366885(j). (similarly for A366886).

Links

Crossrefs

Cf. A000265, A000593, A163511, A347385, A351461, A366887.
Cf. also A324186, A366806, A366885, A366886, A366888.
Differs from A366806 for the first time at n=105, where a(105) = 52, while A366806(105) = 19.

Programs

  • PARI
    up_to = 65537;
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; };
A163511(n) = if(!n, 1, my(p=2, t=1); while(n>1, if(!(n%2), (t*=p), p=nextprime(1+p)); n >>= 1); (t*p));
A206787(n) = sumdiv(n, d, d*issquarefree(2*d));
A336651(n) = { my(f=factor(n>>valuation(n,2))); prod(i=1, #f~, f[i,1]^(f[i,2]-1)); };
A366881aux(n) = [A206787(A163511(n)), A336651(A163511(n))];
v366881 = rgs_transform(vector(1+up_to,n,A366881aux(n-1)));
A366881(n) = v366881[1+n];

Formula

For all n >= 1, a(n) = a(2*n) = a(A000265(n)).

A366798 Lexicographically earliest infinite sequence such that a(i) = a(j) => A366797(i) = A366797(j) for all i, j >= 0, where A366797 is the number of odd divisors permuted by A163511.

Original entry on oeis.org

1, 1, 1, 2, 1, 3, 2, 2, 1, 4, 3, 3, 2, 4, 2, 2, 1, 5, 4, 4, 3, 6, 3, 3, 2, 6, 4, 4, 2, 4, 2, 2, 1, 6, 5, 5, 4, 7, 4, 4, 3, 8, 6, 6, 3, 6, 3, 3, 2, 7, 6, 6, 4, 7, 4, 4, 2, 6, 4, 4, 2, 4, 2, 2, 1, 9, 6, 6, 5, 10, 5, 5, 4, 11, 7, 7, 4, 7, 4, 4, 3, 11, 8, 8, 6, 11, 6, 6, 3, 8, 6, 6, 3, 6, 3, 3, 2, 10, 7, 7, 6, 11, 6, 6
Offset: 0

Views

Author

Antti Karttunen, Oct 27 2023

Keywords

Comments

Restricted growth sequence transform of A366797.

Links

Crossrefs

Cf. A113415, A163511, A366797.
Cf. also A366806, A366874.

Programs

  • PARI
    up_to = 65537;
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; };
A001227(n) = numdiv(n>>valuation(n, 2));
A163511(n) = if(!n, 1, my(p=2, t=1); while(n>1, if(!(n%2), (t*=p), p=nextprime(1+p)); n >>= 1); (t*p));
A366797(n) = A001227(A163511(n));
v366798 = rgs_transform(vector(1+up_to,n,A366797(n-1)));
A366798(n) = v366798[1+n];

A366886 Lexicographically earliest infinite sequence such that a(i) = a(j) => A366885(i) = A366885(j) for all i, j >= 0.

Original entry on oeis.org

1, 1, 1, 2, 1, 3, 2, 4, 1, 5, 3, 6, 2, 7, 4, 8, 1, 9, 5, 10, 3, 11, 6, 12, 2, 13, 7, 14, 4, 15, 8, 3, 1, 16, 9, 17, 5, 18, 10, 19, 3, 20, 11, 21, 6, 22, 12, 23, 2, 24, 13, 25, 7, 26, 14, 27, 4, 27, 15, 13, 8, 14, 3, 28, 1, 29, 16, 30, 9, 31, 17, 32, 5, 33, 18, 34, 10, 35, 19, 36, 3, 37, 20, 38, 11, 39, 21, 40, 6
Offset: 0

Views

Author

Antti Karttunen, Nov 04 2023

Keywords

Comments

Restricted growth sequence transform of A366885.
Albeit quite ugly, the scatter plot is still interesting. - Antti Karttunen, Jan 03 2024

Links

Crossrefs

Cf. A163511, A347385, A366885.
Cf. also A366806, A366881, A366891 (compare the scatter plots).

Programs

  • PARI
    up_to = 65537;
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; };
A163511(n) = if(!n, 1, my(p=2, t=1); while(n>1, if(!(n%2), (t*=p), p=nextprime(1+p)); n >>= 1); (t*p));
A347385(n) = if(1==n,n, my(f=factor(n>>valuation(n, 2))); prod(i=1, #f~, f[i, 1]^f[i, 2] + f[i, 1]^(f[i, 2]-1)));
A366885(n) = A347385(A163511(n));
v366886 = rgs_transform(vector(1+up_to,n,A366885(n-1)));
A366886(n) = v366886[1+n];

A366891 Lexicographically earliest infinite sequence such that a(i) = a(j) => A365425(i) = A365425(j), A206787(A163511(i)) = A206787(A163511(j)) and A336651(A163511(n)) = A336651(A163511(j)) for all i, j >= 0.

Original entry on oeis.org

1, 1, 1, 2, 1, 3, 2, 4, 1, 5, 3, 6, 2, 7, 4, 8, 1, 9, 5, 10, 3, 11, 6, 12, 2, 13, 7, 14, 4, 15, 8, 16, 1, 17, 9, 18, 5, 19, 10, 20, 3, 21, 11, 22, 6, 23, 12, 24, 2, 25, 13, 26, 7, 27, 14, 28, 4, 29, 15, 30, 8, 14, 16, 31, 1, 32, 17, 33, 9, 34, 18, 35, 5, 36, 19, 37, 10, 38, 20, 39, 3, 40, 21, 41, 11, 42, 22, 43, 6, 44, 23
Offset: 0

Views

Author

Antti Karttunen, Nov 04 2023

Keywords

Comments

Restricted growth sequence transform of the triplet [A365425(n), A206787(A163511(n)), A336651(A163511(n))], and also by conjecture, of sequence b(n) = A351040(A163511(n)).
For all i, j >= 0:
a(i) = a(j) => A365395(i) = A365395(j),
a(i) = a(j) => A366874(i) = A366874(j),
a(i) = a(j) => A366881(i) = A366881(j).

Links

Crossrefs

Cf. A000265, A000593, A163511, A347385, A351040, A351461, A365425.
Differs from A366806 for the first time at n=105, where a(105) = 52, while A366806(105) = 19.
Differs from A366881 for the first time at n=511, where a(511) = 249, while A366881(511) = 7.

Programs

  • PARI
    up_to = 65537;
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; };
A000265(n) = (n>>valuation(n,2));
A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); }; \\ From A046523
A163511(n) = if(!n, 1, my(p=2, t=1); while(n>1, if(!(n%2), (t*=p), p=nextprime(1+p)); n >>= 1); (t*p));
A365425(n) = A046523(A000265(A163511(n)));
A206787(n) = sumdiv(n, d, d*issquarefree(2*d));
A336651(n) = { my(f=factor(n>>valuation(n,2))); prod(i=1, #f~, f[i,1]^(f[i,2]-1)); };
A366891aux(n) = [A365425(n), A206787(A163511(n)), A336651(A163511(n))];
v366891 = rgs_transform(vector(1+up_to,n,A366891aux(n-1)));
A366891(n) = v366891[1+n];

Formula

For all n >= 1, a(n) = a(2*n) = a(A000265(n)).

A366893 Lexicographically earliest infinite sequence such that a(i) = a(j) => A366892(i) = A366892(j) for all i, j >= 0.

Original entry on oeis.org

1, 1, 1, 2, 1, 3, 2, 4, 1, 5, 3, 6, 2, 7, 4, 8, 1, 9, 5, 10, 3, 11, 6, 12, 2, 13, 7, 14, 4, 15, 8, 16, 1, 17, 9, 18, 5, 19, 10, 20, 3, 21, 11, 22, 6, 23, 12, 24, 2, 25, 13, 26, 7, 27, 14, 28, 4, 29, 15, 30, 8, 31, 16, 32, 1, 33, 17, 34, 9, 35, 18, 36, 5, 37, 19, 38, 10, 39, 20, 40, 3, 41, 21, 42, 11, 43, 22, 44, 6, 45, 23
Offset: 0

Views

Author

Antti Karttunen, Nov 04 2023

Keywords

Comments

Restricted growth sequence transform of A366892.

Links

Crossrefs

Cf. A163511, A336652, A366892.
Cf. also A366806.
Differs from A003602 for the first time at n=121, where a(121) = 24, while A003602(121) = 61.

Programs

  • PARI
    up_to = 65537;
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; };
A163511(n) = if(!n, 1, my(p=2, t=1); while(n>1, if(!(n%2), (t*=p), p=nextprime(1+p)); n >>= 1); (t*p));
A336652(n) = if(1==n,n,my(f=factor(n)); prod(i=1,#f~,if(2==f[i,1],1,-1+(((f[i,1]^(1+f[i,2]))-1) / (f[i,1]-1)))));
A366892(n) = A336652(A163511(n));
v366893 = rgs_transform(vector(1+up_to,n,A366892(n-1)));
A366893(n) = v366893[1+n];

A366895 Lexicographically earliest infinite sequence such that a(i) = a(j) => A366894(i) = A366894(j) for all i, j >= 0.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 2, 1, 1, 1, 1, 1, 1, 4, 3, 5, 2, 1, 1, 6, 1, 5, 1, 1, 1, 1, 1, 1, 1, 7, 4, 8, 3, 5, 5, 9, 2, 10, 1, 11, 1, 6, 6, 12, 1, 1, 5, 1, 1, 1, 1, 1, 1, 2, 1, 5, 1, 1, 1, 1, 1, 13, 7, 14, 4, 8, 8, 15, 3, 16, 5, 17, 5, 9, 9, 7, 2, 18, 10, 19, 1, 11, 11, 12, 1, 20, 6, 21, 6, 12, 12, 7, 1, 22, 1, 5, 5
Offset: 0

Views

Author

Antti Karttunen, Jan 03 2024

Keywords

Comments

Restricted growth sequence transform of A366894.
For all i, j >= 0:
A366881(i) = A366881(j) => A366806(i) = A366806(j) => a(i) = a(j).

Links

Crossrefs

Cf. A000265, A000593, A163511, A324186, A336699, A351565, A366894.
Cf. also A366806, A366881.

Programs

  • PARI
    up_to = 65537;
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; };
A000265(n) = (n>>valuation(n,2));
A163511(n) = if(!n,1,my(p=2, t=1); while(n>1, if(!(n%2), (t*=p), p=nextprime(1+p)); n >>= 1); (t*p));
A336699(n) = A000265(1+A000265(sigma(A000265(n))));
A366894(n) = A336699(A163511(n));
v366895 = rgs_transform(vector(1+up_to,n,A366894(n-1)));
A366895(n) = v366895[1+n];
Showing 1-7 of 7 results.

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