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 1-4 of 4 results.

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];

A366873 a(n) = A113415(A163511(n)), where A113415(n) is the average of number of and sum of odd divisors of n.

Original entry on oeis.org

1, 1, 1, 3, 1, 8, 3, 4, 1, 22, 8, 17, 3, 14, 4, 5, 1, 63, 22, 80, 8, 65, 17, 30, 3, 42, 14, 26, 4, 18, 5, 7, 1, 185, 63, 393, 22, 316, 80, 202, 8, 206, 65, 174, 17, 117, 30, 68, 3, 124, 42, 127, 14, 100, 26, 50, 4, 55, 18, 38, 5, 26, 7, 8, 1, 550, 185, 1956, 63, 1567, 393, 1403, 22, 1020, 316, 1204, 80, 804, 202
Offset: 0

Views

Author

Antti Karttunen, Oct 27 2023

Keywords

Links

Crossrefs

Cf. A113415, A163511, A366874 (rgs-transform).
Cf. also A324186, A366797, A366875.

Programs

  • PARI
    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));

Formula

a(n) = (1/2) * (A324186(n)+A366797(n)).

A366878 Lexicographically earliest infinite sequence such that a(i) = a(j) => A366875(i) = A366875(j) for all i, j >= 0.

Original entry on oeis.org

1, 2, 3, 4, 3, 1, 5, 6, 3, 2, 2, 2, 3, 7, 8, 9, 3, 10, 1, 11, 3, 12, 1, 9, 3, 10, 13, 14, 3, 15, 12, 16, 3, 6, 17, 18, 3, 19, 20, 21, 3, 1, 9, 22, 3, 23, 12, 24, 3, 25, 17, 26, 3, 27, 28, 29, 3, 10, 30, 23, 3, 31, 32, 11, 3, 11, 8, 33, 3, 34, 35, 36, 3, 11, 37, 38, 3, 39, 40, 41, 3, 5, 2, 42, 3, 43, 44, 45, 3, 1, 46, 47, 3
Offset: 0

Views

Author

Antti Karttunen, Oct 27 2023

Keywords

Comments

Restricted growth sequence transform of A366875.

Links

Crossrefs

Cf. A113415, A163511, A349915, A366875.
Cf. also A366383, 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; };
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));
A113415(n) = if(n<1, 0, sumdiv(n, d, if(d%2, (d+1)/2)));
memoA349915 = Map();
A349915(n) = if(1==n,1,my(v); if(mapisdefined(memoA349915,n,&v), v, v = -sumdiv(n,d,if(dA113415(n/d)*A349915(d),0)); mapput(memoA349915,n,v); (v)));
A366875(n) = A349915(A163511(n));
v366878 = rgs_transform(vector(1+up_to,n,A366875(n-1)));
A366878(n) = v366878[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)).
Showing 1-4 of 4 results.

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