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.

Previous Showing 31-40 of 61 results. Next

A331747 Lexicographically earliest infinite sequence such that a(i) = a(j) => A009194(i) = A009194(j) and A278222(i) = A278222(j) for all i, j.

Original entry on oeis.org

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

Views

Author

Antti Karttunen, Feb 04 2020

Keywords

Comments

Restricted growth sequence transform of the ordered pair [A009194(n), A278222(n)].
For all i, j:
A331746(i) = A331746(j) => a(i) = a(j).

Links

Crossrefs

Cf. A009194, A278222.
Cf. also A324400, A286622, A318310, A318311, A324389, A331744, A331745, A331746.

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; };
A009194(n) = gcd(n, sigma(n));
A005940(n) = { my(p=2, t=1); n--; until(!n\=2, if((n%2), (t*=p), p=nextprime(p+1)));
t };
A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); };  \\ From A046523
A278222(n) = A046523(A005940(1+n));
Aux331747(n) = [A009194(n),A278222(n)];
v331747 = rgs_transform(vector(up_to, n, Aux331747(n)));
A331747(n) = v331747[n];

Formula

a(2^n) = 1 for all n >= 0.

A336162 Lexicographically earliest infinite sequence such that a(i) = a(j) => A278222(i) = A278222(j) and A335915(i) = A335915(j), for all i, j >= 1.

Original entry on oeis.org

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

Views

Author

Antti Karttunen, Jul 11 2020

Keywords

Comments

Restricted growth sequence transform of the ordered pair [A278222(n), A335915(n)].
For all i, j: A324400(i) = A324400(j) => a(i) = a(j).

Links

Crossrefs

Cf. A000265, A005940, A046523, A278222, A335915.
Cf. also A286622, A324400, A336155, A336159, A336160.

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));
A005940(n) = { my(p=2, t=1); n--; until(!n\=2, if((n%2), (t*=p), p=nextprime(p+1))); (t); };
A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); };  \\ From A046523
A278222(n) = A046523(A005940(1+n));
A335915(n) = { my(f=factor(n)); prod(k=1,#f~,if(2==f[k,1],1,(A000265(f[k,1]-1)*A000265(f[k,1]+1))^f[k,2])); };
Aux336162(n) = [A278222(n), A335915(n)];
v336162 = rgs_transform(vector(up_to, n, Aux336162(n)));
A336162(n) = v336162[n];

A336935 Lexicographically earliest infinite sequence such that a(i) = a(j) => A007733(i) = A007733(j) and A278222(i) = A278222(j), for all i, j >= 1.

Original entry on oeis.org

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

Views

Author

Antti Karttunen, Aug 10 2020

Keywords

Comments

Restricted growth sequence transform of the ordered pair [A007733(n), A278222(n)].
For all i, j: A324400(i) = A324400(j) => A003602(i) = A003602(j) => a(i) = a(j).

Links

Crossrefs

Cf. A000265, A003602, A007733, A278222, A286622, A324400, A336933, A336934.

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; };
A007733(n) = znorder(Mod(2, n/2^valuation(n, 2))); \\ From A007733
A005940(n) = { my(p=2, t=1); n--; until(!n\=2, if((n%2), (t*=p), p=nextprime(p+1))); t };
A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); };  \\ From A046523
A278222(n) = A046523(A005940(1+n));
Aux336935(n) = [A007733(n), A278222(n)];
v336935 = rgs_transform(vector(up_to, n, Aux336935(n)));
A336935(n) = v336935[n];

A351963 Lexicographically earliest infinite sequence such that a(i) = a(j) => A278222(A109812(i)) = A278222(A109812(j)), for all i, j >= 1.

Original entry on oeis.org

1, 1, 1, 2, 1, 3, 3, 1, 2, 3, 3, 2, 3, 4, 1, 4, 2, 3, 3, 5, 3, 5, 3, 6, 3, 5, 2, 7, 1, 5, 6, 3, 5, 5, 3, 5, 6, 3, 8, 2, 9, 3, 5, 4, 5, 5, 3, 8, 5, 1, 8, 5, 3, 7, 5, 3, 10, 6, 3, 6, 6, 5, 6, 5, 6, 5, 5, 4, 8, 3, 9, 5, 5, 7, 2, 11, 3, 10, 3, 10, 6, 6, 9, 5, 8, 6, 5, 8, 5, 10, 6, 12, 6, 10, 6, 5, 10, 4, 6, 8, 5, 13
Offset: 1

Views

Author

Antti Karttunen, Apr 06 2022

Keywords

Comments

Restricted growth sequence transform of A278222(A109812(n)), or equally of, A278222(A351965(n)).
For all i, j: A351578(i) = A351578(j) => a(i) = a(j) => A352884(i) = A352884(j).

Links

Crossrefs

Cf. A005940, A046523, A109812, A278222, A286622, A351965, A352884.
Cf. also A351578, A352888.

Programs

  • PARI
    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; };
v109812 = readvec("b109812_to10e5.txt"); \\ Prepared from b-file data with gawk ' { print $2 } '
up_to = #v109812;
A109812(n) = v109812[n];
A005940(n) = { my(p=2, t=1); n--; until(!n\=2, if((n%2), (t*=p), p=nextprime(p+1))); (t); };
A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); }; \\ From A046523
v351963 = rgs_transform(vector(up_to, n, A046523(A005940(1+A109812(n)))));
A351963(n) = v351963[n];

A286581 Restricted growth sequence transform of A286580.

Original entry on oeis.org

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

Views

Author

Antti Karttunen, Jun 03 2017

Keywords

Links

Crossrefs

Cf. A233275, A286580, A286537, A286539, A286617, A286619, A286622.

A286589 Restricted growth sequence transform of A286588.

Original entry on oeis.org

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

Views

Author

Antti Karttunen, Jun 03 2017

Keywords

Links

Crossrefs

Cf. A286588, A286622.

A323234 Lexicographically earliest sequence such that a(i) = a(j) => f(i) = f(j), where f(1) = 0, and for n > 1, f(n) = ordered pair [A053645(n), A079944(n-2)], where A053645(n) gives n without its most significant bit, while A079944(n-2) gives the second most significant bit of n.

Original entry on oeis.org

1, 2, 3, 2, 4, 5, 6, 2, 4, 7, 8, 9, 10, 11, 12, 2, 4, 7, 8, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 2, 4, 7, 8, 13, 14, 15, 16, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 2, 4, 7, 8, 13, 14, 15, 16, 25, 26, 27, 28, 29, 30, 31, 32, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61
Offset: 1

Views

Author

Antti Karttunen, Jan 08 2019

Keywords

Comments

Also the restricted growth sequence transform of function f(1) = 0, f(n) = [A053645(n), A278222(n)] for n > 1.
For all i, j:
a(i) = a(j) => A286622(i) = A286622(j),
a(i) = a(j) => A323235(i) = A323235(j),
a(i) = a(j) => A323236(i) = A323236(j).

Links

Crossrefs

Cf. A053645, A079944, A278222, A286622, A323235, A323236.
Cf. also A300226 (an analogous filter sequence for prime factorization).

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; };
A053644(n) = { my(k=1); while(k<=n, k<<=1); (k>>1); }; \\ From A053644
A053645(n) = (n-A053644(n));
A079944off0(n) = (1==binary(2+n)[2]);
A323234aux(n) = if(1==n,0,[A053645(n), A079944off0(n-2)]);
v323234 = rgs_transform(vector(up_to,n,A323234aux(n)));
A323234(n) = v323234[n];

A324345 Lexicographically earliest positive sequence such that a(i) = a(j) => A005811(i) = A005811(j) and A278222(i) = A278222(j), for all i, j >= 0.

Original entry on oeis.org

1, 2, 3, 4, 3, 5, 6, 7, 3, 5, 8, 9, 6, 9, 10, 11, 3, 5, 8, 9, 8, 12, 13, 14, 6, 9, 13, 15, 10, 14, 16, 17, 3, 5, 8, 9, 8, 12, 13, 14, 8, 12, 18, 19, 13, 19, 20, 21, 6, 9, 13, 15, 13, 19, 22, 23, 10, 14, 20, 23, 16, 21, 24, 25, 3, 5, 8, 9, 8, 12, 13, 14, 8, 12, 18, 19, 13, 19, 20, 21, 8, 12, 18, 19, 18, 26, 27, 28, 13, 19, 27, 29, 20, 28, 30, 31, 6, 9, 13, 15, 13, 19
Offset: 0

Views

Author

Antti Karttunen, Feb 24 2019

Keywords

Comments

Restricted growth sequence transform of the ordered pair [A005811(n), A278222(n)], or equally, of [A005811(n), A286622(n)].
For all i, j >= 1:
a(i) = a(j) => A033264(i) = A033264(j).

Links

Crossrefs

Cf. A005811, A278222, A286622.
Cf. also A033264, A323889, A324343, A324346.

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; };
A005811(n) = hammingweight(bitxor(n, n>>1)); \\ From A005811
A005940(n) = { my(p=2, t=1); n--; until(!n\=2, if((n%2), (t*=p), p=nextprime(p+1))); t };
A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); };  \\ From A046523
A278222(n) = A046523(A005940(1+n));
Aux324345(n) = [A005811(n), A278222(n)];
v324345 = rgs_transform(vector(1+up_to,n,Aux324345(n-1)));
A324345(n) = v324345[1+n];

Formula

a(2^n) = 3 for all n >= 1.

A336154 Lexicographically earliest infinite sequence such that a(i) = a(j) => A007814(1+i) = A007814(1+j) and A278222(i) = A278222(j), for all i, j >= 1.

Original entry on oeis.org

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

Views

Author

Antti Karttunen, Jul 11 2020

Keywords

Comments

Restricted growth sequence transform of the ordered pair [A007814(1+n), A278222(n)]. Note that A007814(1+n) gives the number of trailing 1-bits in the binary expansion of n.
For all i, j: A324400(i) = A324400(j) => a(i) = a(j).

Links

Crossrefs

Cf. A007814, A278222.
Cf. also A286622, A324400, A336153, A336155, A336156.

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; };
A007814(n) = valuation(n,2);
A005940(n) = { my(p=2, t=1); n--; until(!n\=2, if((n%2), (t*=p), p=nextprime(p+1))); (t); };
A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); };  \\ From A046523
A278222(n) = A046523(A005940(1+n));
Aux336154(n) = [A007814(1+n), A278222(n)];
v336154 = rgs_transform(vector(up_to, n, Aux336154(n)));
A336154(n) = v336154[n];

A336394 Lexicographically earliest infinite sequence such that a(i) = a(j) => A278222(i) = A278222(j) and A331410(i) = A331410(j), for all i, j >= 1.

Original entry on oeis.org

1, 1, 2, 1, 3, 2, 4, 1, 3, 3, 5, 2, 5, 4, 6, 1, 7, 3, 8, 3, 9, 5, 10, 2, 11, 5, 12, 4, 13, 6, 14, 1, 7, 7, 8, 3, 15, 8, 16, 3, 17, 9, 18, 5, 19, 10, 20, 2, 5, 11, 21, 5, 19, 12, 22, 4, 13, 13, 22, 6, 20, 14, 23, 1, 24, 7, 11, 7, 17, 8, 16, 3, 25, 15, 26, 8, 18, 16, 27, 3, 15, 17, 18, 9, 28, 18, 29, 5, 26, 19, 30, 10, 31, 20, 32, 2, 8, 5, 21, 11
Offset: 1

Views

Author

Antti Karttunen, Aug 10 2020

Keywords

Comments

Restricted growth sequence transform of the ordered pair [A278222(n), A331410(n)].
For all i, j: A324400(i) = A324400(j) => A003602(i) = A003602(j) => a(i) = a(j).

Links

Crossrefs

Cf. A003602, A278222, A324400, A331410, A336392.
Cf. also A286622, A336473.

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; };
A005940(n) = { my(p=2, t=1); n--; until(!n\=2, if((n%2), (t*=p), p=nextprime(p+1))); t };
A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); };  \\ From A046523
A278222(n) = A046523(A005940(1+n));
A331410(n) = if(!bitand(n,n-1),0,1+A331410(n+(n/vecmax(factor(n)[, 1]))));
Aux336394(n) = [A278222(n), A331410(n)];
v336394 = rgs_transform(vector(up_to, n, Aux336394(n)));
A336394(n) = v336394[n];
Previous Showing 31-40 of 61 results. Next

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.