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.

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A336161 Lexicographically earliest infinite sequence such that a(i) = a(j) => A087436(i) = A087436(j) and A335915(i) = A335915(j) for all i, j >= 1.

Original entry on oeis.org

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

Views

Author

Antti Karttunen, Jul 10 2020

Keywords

Comments

Restricted growth sequence transform of the ordered pair [A087436(n), A335915(n)].
For all i, j: A324400(i) = A324400(j) => a(i) = a(j) => A335904(i) = A335904(j).

Links

Crossrefs

Cf. A087436, A335915.
Cf. also A324400, A335880, A335904.

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; };
A087436(n) = (bigomega(n>>valuation(n,2)));
A000265(n) = (n>>valuation(n,2));
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])); };
Aux336161(n) = [A087436(n),A335915(n)];
v336161 = rgs_transform(vector(up_to, n, Aux336161(n)));
A336161(n) = v336161[n];

A336392 Lexicographically earliest infinite sequence such that a(i) = a(j) => A278222(i) = A278222(j) and A336467(i) = A336467(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, 3, 9, 6, 5, 17, 10, 18, 3, 19, 11, 20, 6, 21, 12, 22, 2, 23, 13, 24, 7, 25, 14, 26, 4, 27, 15, 28, 8, 29, 16, 30, 1, 31, 3, 32, 9, 33, 6, 34, 5, 35, 17, 36, 10, 21, 18, 37, 3, 11, 19, 38, 11, 39, 20, 40, 6, 41, 21, 42, 12, 43, 22, 44, 2, 45, 23, 46, 13
Offset: 1

Views

Author

Antti Karttunen, Aug 10 2020

Keywords

Comments

Restricted growth sequence transform of the ordered pair [A278222(n), A336467(n)].
For all i, j: A324400(i) = A324400(j) => A003602(i) = A003602(j) => a(i) = a(j).

Links

Crossrefs

Cf. A003602, A278222, A324400, A336390, A336391, A336394, A336467.
Cf. also A286622, A336472.

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));
A000265(n) = (n>>valuation(n,2));
A336467(n) = { my(f=factor(n)); prod(k=1,#f~,if(2==f[k,1],1,(A000265(f[k,1]+1))^f[k,2])); };
Aux336392(n) = [A278222(n), A336467(n)];
v336392 = rgs_transform(vector(up_to, n, Aux336392(n)));
A336392(n) = v336392[n];

A336393 Lexicographically earliest infinite sequence such that a(i) = a(j) => A336467(i) = A336467(j) and A278221(A000265(i)) = A278221(A000265(j)), for all i, j >= 1.

Original entry on oeis.org

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

Views

Author

Antti Karttunen, Aug 10 2020

Keywords

Comments

Restricted growth sequence transform of the ordered pair [A336467(n), A278221(A000265(n))], or equally, of the ordered pair [A336467(n), A336395(n)].
For all i, j:
A324400(i) = A324400(j) => A003602(i) = A003602(j) => a(i) = a(j).

Links

Crossrefs

Cf. A000265, A003602, A324400, A278221, A286621, A336390, A336395, A336467.

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
A122111(n) = if(1==n,n,my(f=factor(n), es=Vecrev(f[,2]),is=concat(apply(primepi,Vecrev(f[,1])),[0]),pri=0,m=1); for(i=1, #es, pri += es[i]; m *= prime(pri)^(is[i]-is[1+i])); (m));
A278221(n) = A046523(A122111(n));
A336467(n) = { my(f=factor(n)); prod(k=1,#f~,if(2==f[k,1],1,(A000265(f[k,1]+1))^f[k,2])); };
Aux336393(n) = [A336467(n), A278221(A000265(n))];
v336393 = rgs_transform(vector(up_to, n, Aux336393(n)));
A336393(n) = v336393[n];

A336926 Lexicographically earliest infinite sequence such that a(i) = a(j) => A335880(1+sigma(i)) = A335880(1+sigma(j)), for all i, j >= 1.

Original entry on oeis.org

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

Views

Author

Antti Karttunen, Aug 10 2020

Keywords

Comments

Restricted growth sequence transform of the function f(n) = A335880(A088580(n)).
For all i, j:
A324400(i) = A324400(j) => a(i) = a(j),
a(i) = a(j) => A336694(i) = A336694(j),
a(i) = a(j) => A336695(i) = A336695(j).

Links

Crossrefs

Cf. A000203, A088580, A329697, A331410, A332459, A335880, A336694, A336695.
Cf. also A336925.

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; };
A329697(n) = { my(f=factor(n)); sum(k=1,#f~,if(2==f[k,1],0,f[k,2]*(1+A329697(f[k,1]-1)))); };
A331410(n) = { my(f=factor(n)); sum(k=1,#f~,if(2==f[k,1],0,f[k,2]*(1+A331410(f[k,1]+1)))); };
Aux335880(n) = [A329697(n),A331410(n)];
v336926 = rgs_transform(vector(up_to, n, Aux335880(1+sigma(n))));
A336926(n) = v336926[n];

A351453 Lexicographically earliest infinite sequence such that a(i) = a(j) => A006530(i) = A006530(j) and A007733(i) = A007733(j) for all i, j >= 1.

Original entry on oeis.org

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

Views

Author

Antti Karttunen, Feb 11 2022

Keywords

Comments

Restricted growth sequence transform of the ordered pair [A006530(n), A007733(n)].
For all i, j >= 1: A324400(i) = A324400(j) => a(i) = a(j).

Links

Crossrefs

Cf. A006530, A007733.
Cf. also A324400, A351452, A351454, A351460.

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; };
A006530(n) = if(1==n, n, my(f=factor(n)); f[#f~, 1]);
A007733(n) = znorder(Mod(2, n/2^valuation(n, 2))); \\ This function from A007733
Aux351453(n) = [A006530(n), A007733(n)];
v351453 = rgs_transform(vector(up_to, n, Aux351453(n)));
A351453(n) = v351453[n];

A324533 Lexicographically earliest positive sequence such that a(i) = a(j) => A002487(i) = A002487(j) and A278219(i) = A278219(j), for all i, j >= 0.

Original entry on oeis.org

1, 2, 3, 4, 3, 5, 6, 7, 3, 8, 9, 10, 6, 11, 12, 13, 3, 11, 14, 15, 9, 16, 17, 18, 6, 19, 17, 20, 12, 15, 21, 22, 3, 23, 24, 25, 14, 26, 27, 28, 9, 29, 30, 31, 17, 32, 33, 34, 6, 35, 27, 36, 17, 37, 38, 39, 12, 40, 33, 39, 21, 25, 41, 42, 3, 15, 43, 39, 24, 44, 45, 46, 14, 47, 48, 49, 27, 50, 51, 46, 9, 52, 48, 53, 30, 54, 55, 56, 17, 57, 58, 59, 33, 60, 61, 62, 6
Offset: 0

Views

Author

Antti Karttunen, Mar 05 2019

Keywords

Comments

Restricted growth sequence transform of the ordered pair [A002487(n), A278219(n)].

Links

Crossrefs

Cf. A002487, A278219, A286619, A324400.
Cf. also A323889 (compare the scatterplots).

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; };
A002487(n) = { my(a=1, b=0); while(n>0, if(bitand(n, 1), b+=a, a+=b); n>>=1); (b); }; \\ From A002487
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
A003188(n) = bitxor(n, n>>1);
A278219(n) = A046523(A005940(1+A003188(n)));
Aux324533(n) = [A002487(n), A278219(n)];
v324533 = rgs_transform(vector(1+up_to,n,Aux324533(n-1)));
A324533(n) = v324533[1+n];

Formula

For n >= 1, a(2^n) = 3.

A325810 Lexicographically earliest sequence such that a(i) = a(j) => A034460(i) = A034460(j) and A325814(i) = A325814(j) for all i, j.

Original entry on oeis.org

1, 2, 3, 2, 4, 5, 6, 2, 7, 8, 9, 10, 11, 12, 13, 2, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 2, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 35, 2, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 45, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87
Offset: 1

Views

Author

Antti Karttunen, May 23 2019

Keywords

Comments

Restricted growth sequence transform of the ordered pair [A034460(n), A325814(n)].
For all i, j:
A324400(i) = A324400(j) => a(i) = a(j),
a(i) = a(j) => A033879(i) = A033879(j),
a(i) = a(j) => A325811(i) = A325811(j) => A325813(i) = A325813(i).

Links

Crossrefs

Cf. A033879, A034460, A324400, A325811, A325813, A325814.

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; };
A034448(n) = { my(f=factorint(n)); prod(k=1, #f~, 1+(f[k, 1]^f[k, 2])); }; \\ After code in A034448
A034460(n) = (A034448(n) - n);
A048146(n) = (sigma(n)-A034448(n));
A325814(n) = (n-A048146(n));
v325810 = rgs_transform(vector(up_to,n,[A034460(n), A325814(n)]));
A325810(n) = v325810[n];

A332900 Lexicographically earliest infinite sequence such that a(i) = a(j) => f(i) = f(j), where f(n) = 0 if n > 1 and n is a square or twice square, with f(n) = n for all other numbers.

Original entry on oeis.org

1, 2, 3, 2, 4, 5, 6, 2, 2, 7, 8, 9, 10, 11, 12, 2, 13, 2, 14, 15, 16, 17, 18, 19, 2, 20, 21, 22, 23, 24, 25, 2, 26, 27, 28, 2, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 2, 2, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 2, 54, 55, 56, 57, 58, 59, 60, 2, 61, 62, 63, 64, 65, 66, 67, 68, 2, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 2, 85, 2
Offset: 1

Views

Author

Antti Karttunen, Mar 05 2020

Keywords

Comments

For all i, j:
A324400(i) = A324400(j) => a(i) = a(j),
a(i) = a(j) => A292383(i) = A292383(j) => A292583(i) = A292583(j),
a(i) = a(j) => A332896(i) = A332896(j) => A332901(i) = A332901(j).

Links

Crossrefs

Cf. A292383, A292583, A324400, A332896, A332901.

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; };
A332900aux(n) = if((n>1)&&(issquare(n)||issquare(2*n)),0,n);
v332900 = rgs_transform(vector(up_to,n,A332900aux(n)));
A332900(n) = v332900[n];

A336395 Lexicographically earliest infinite sequence such that a(i) = a(j) => A278221(A000265(i)) = A278221(A000265(j)), for all i, j >= 1.

Original entry on oeis.org

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

Views

Author

Antti Karttunen, Aug 10 2020

Keywords

Comments

Restricted growth sequence transform of the function f(n) = A278221(A000265(n)), the prime signature of the conjugated prime factorization of the odd part of n.
For all i, j:
A324400(i) = A324400(j) => A003602(i) = A003602(j) => a(i) = a(j),
a(i) = a(j) => A005087(i) = A005087(j).

Links

Crossrefs

Cf. A000265, A003602, A005087, A278221, A286621, A324400.
Cf. also A336393.

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
A122111(n) = if(1==n,n,my(f=factor(n), es=Vecrev(f[,2]),is=concat(apply(primepi,Vecrev(f[,1])),[0]),pri=0,m=1); for(i=1, #es, pri += es[i]; m *= prime(pri)^(is[i]-is[1+i])); (m));
A278221(n) = A046523(A122111(n));
v336395 = rgs_transform(vector(up_to, n, A278221(A000265(n))));
A336395(n) = v336395[n];

A336474 Lexicographically earliest infinite sequence such that a(i) = a(j) => A278221(i) = A278221(j) and A329697(i) = A329697(j), for all i, j >= 1.

Original entry on oeis.org

1, 2, 3, 2, 4, 5, 6, 2, 7, 8, 9, 5, 10, 11, 12, 2, 13, 14, 15, 8, 16, 17, 18, 5, 19, 20, 21, 11, 22, 23, 24, 2, 25, 26, 27, 14, 28, 29, 30, 8, 31, 32, 33, 17, 34, 35, 36, 5, 37, 12, 38, 20, 39, 40, 25, 11, 41, 42, 43, 23, 44, 45, 46, 2, 47, 48, 49, 26, 50, 32, 51, 14, 52, 53, 34, 29, 54, 55, 56, 8, 57, 58, 59, 32, 60, 61, 62, 17, 63, 64, 65, 35, 66, 67, 68, 5, 69, 70, 71, 12, 72, 73, 74, 20, 75
Offset: 1

Views

Author

Antti Karttunen, Jul 25 2020

Keywords

Comments

Restricted growth sequence transform of the ordered pair [A278221(n), A329697(n)].
For all i, j: A324400(i) = A324400(j) => A336146(i) = A336146(j) => a(i) = a(j).

Links

Crossrefs

Cf. A046523, A122111, A278221, A329697.
Cf. also A324400, A336146, 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; };
A122111(n) = if(1==n,n,my(f=factor(n), es=Vecrev(f[,2]),is=concat(apply(primepi,Vecrev(f[,1])),[0]),pri=0,m=1); for(i=1, #es, pri += es[i]; m *= prime(pri)^(is[i]-is[1+i])); (m));
A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); }; \\ From A046523
A278221(n) = A046523(A122111(n));
A329697(n) = if(!bitand(n,n-1),0,1+A329697(n-(n/vecmax(factor(n)[, 1]))));
Aux336474(n) = [A278221(n), A329697(n)];
v336474 = rgs_transform(vector(up_to,n,Aux336474(n)));
A336474(n) = v336474[n];
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