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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A335421 Lexicographically earliest infinite sequence such that a(i) = a(j) => A046523(A335422(i)) = A046523(A335422(j)) for all i, j >= 0.

Original entry on oeis.org

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

Views

Author

Antti Karttunen, Jun 09 2020

Keywords

Comments

For all i, j: A324400(i) = A324400(j) => a(i) = a(j) => A335420(i) = A335420(j).

Links

Crossrefs

Cf. A046523, A335420, A335422.
Cf. also A286622, A324400.

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
A054429(n) = ((3<<#binary(n\2))-n-1); \\ From A054429
A163511(n) = if(!n,1,A005940(1+A054429(n)))
A335422(n) = A005940(1+A163511(n));
v335421 = rgs_transform(vector(1+up_to,n,A046523(A335422(n-1))));
A335421(n) = v335421[1+n];

Formula

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

A335424 Lexicographically earliest infinite sequence such that a(i) = a(j) => A046523(A335423(i)) = A046523(A335423(j)) for all i, j >= 1.

Original entry on oeis.org

1, 2, 2, 1, 2, 3, 2, 2, 1, 4, 2, 2, 2, 4, 3, 1, 2, 2, 2, 2, 4, 4, 2, 3, 1, 4, 2, 2, 2, 5, 2, 2, 4, 4, 3, 1, 2, 4, 4, 4, 2, 6, 2, 2, 2, 4, 2, 2, 1, 2, 4, 2, 2, 3, 4, 4, 4, 4, 2, 3, 2, 4, 2, 1, 4, 6, 2, 2, 4, 6, 2, 2, 2, 4, 2, 2, 3, 6, 2, 2, 1, 4, 2, 4, 4, 4, 4, 4, 2, 4, 4, 2, 4, 4, 4, 3, 2, 2, 2, 1, 2, 6, 2, 4, 5
Offset: 1

Views

Author

Antti Karttunen, Jun 13 2020

Keywords

Comments

For all i, j: A305800(i) = A305800(j) => a(i) = a(j) => A162642(i) = A162642(j).

Links

Crossrefs

Cf. A046523, A162642, A248663, A335423, A335425.
Cf. also A286622, A305800.

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; };
A048675(n) = { my(f = factor(n)); sum(k=1, #f~, f[k, 2]*2^primepi(f[k, 1]))/2; };
A248663(n) = A048675(core(n));
A005940(n) = { my(p=2, t=1); n--; until(!n\=2, if((n%2), (t*=p), p=nextprime(p+1))); t };
A335423(n) = A005940(1+A248663(n));
A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); };
v335424 = rgs_transform(vector(up_to,n,A046523(A335423(n))));
A335424(n) = v335424[n];

A336320 Lexicographically earliest infinite sequence such that a(i) = a(j) => A324058(i) = A324058(j) for all i, j >= 0.

Original entry on oeis.org

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

Views

Author

Antti Karttunen, Jul 19 2020

Keywords

Comments

Restricted growth sequence transform of A324058.

Links

Crossrefs

Cf. A005940, A324121, A324058.
Cf. also A286622.

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 }; \\ From A005940
A324121(n) = gcd(sigma(n),n*numdiv(n));
A324058(n) = A324121(A005940(1+n));
v336320 = rgs_transform(vector(1+up_to,n,A324058(n-1)));
A336320(n) = v336320[1+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];

A365720 Lexicographically earliest infinite sequence such that a(i) = a(j) => A365719(i) = A365719(j) for all i, j >= 0, where A365719(n) = A046523(A356867(1+n)).

Original entry on oeis.org

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

Views

Author

Antti Karttunen, Sep 17 2023

Keywords

Comments

Restricted growth sequence transform of A365719.
For all i, j >= 0:
A365718(i) = A365718(j) => a(i) = a(j),
a(i) = a(j) => A365721(i) = A365721(j),
a(i) = a(j) => A365722(i) = A365722(j).

Links

Crossrefs

Cf. A046523, A356867, A365718, A365720 (rgs-transform), A365721, A365722.
Cf. also A286622.

Programs

  • PARI
    up_to = 59049; \\ = 3^10.
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; };
A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); };
A356867list(up_to) = { my(v=vector(up_to),met=Map(),h=0,ak); for(i=1,#v,if(1==vecsum(digits(i,3)), v[i] = i; h = i, ak = v[i-h]; forprime(p=2,,if(3!=p && !mapisdefined(met,p*ak), v[i] = p*ak; break))); mapput(met,v[i],i)); (v); };
v365720 = rgs_transform(apply(A046523,A356867list(1+up_to)));
A365720(n) = v365720[1+n];

A286614 Restricted growth sequence transform of A286613.

Original entry on oeis.org

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

Views

Author

Antti Karttunen, May 30 2017

Keywords

Links

Crossrefs

Cf. A286613, A286622.

Programs

  • PARI
    rgs_transform(invec) = { my(occurrences = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(occurrences,invec[i]), my(pp = mapget(occurrences, invec[i])); outvec[i] = outvec[pp] , mapput(occurrences,invec[i],i); outvec[i] = u; u++ )); outvec; };
write_to_bfile(start_offset,vec,bfilename) = { for(n=1, length(vec), write(bfilename, (n+start_offset)-1, " ", vec[n])); }
A003961(n) = my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); \\ This function from Michel Marcus
A048673(n) = (A003961(n)+1)/2;
A005940(n) = { my(p=2, t=1); n--; until(!n\=2, if((n%2), (t*=p), p=nextprime(p+1))); t }; \\ Modified from code of M. F. Hasler
A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); };  \\ This function from Charles R Greathouse IV, Aug 17 2011
A286613(n) = A046523(A048673(A005940(1+n)));
write_to_bfile(0,rgs_transform(vector(65537,n,A286613(n-1))),"b286614.txt");

A304738 Restricted growth sequence transform of A278222(A048673(n)).

Original entry on oeis.org

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

Views

Author

Antti Karttunen, May 18 2018

Keywords

Comments

Sequence allots a distinct value for each distinct multiset formed from the lengths of 1-runs in the binary representation of A048673(n). Compare to the scatter plot of A286622.

Links

Crossrefs

Cf. A003961, A048673, A278222, A286622.
Cf. also A286243, A304101, A304737.

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; };
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));
A003961(n) = my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); \\ From A003961
A048673(n) = (A003961(n)+1)/2;
v304738 = rgs_transform(vector(65539,n,A278222(A048673(n))));
A304738(n) = v304738[n];

A304744 Restricted growth sequence transform of A046523(A052330(n)).

Original entry on oeis.org

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

Views

Author

Antti Karttunen, May 27 2018

Keywords

Comments

For all i, j: a(i) = a(j) => A000120(i) = A000120(j).

Links

Crossrefs

Cf. A046523, A052330.
Cf. also A286622 (A278222), A302791, A304535, A304745.

Programs

  • PARI
    up_to_e = 17; \\ Good for computing up to n = (2^up_to_e)-1
v050376 = vector(up_to_e);
ispow2(n) = (n && !bitand(n,n-1));
i = 0; for(n=1,oo,if(ispow2(isprimepower(n)), i++; v050376[i] = n); if(i == up_to_e,break));
A050376(n) = v050376[n];
A052330(n) = { my(p=1,i=1); while(n>0, if(n%2, p *= A050376(i)); i++; n >>= 1); (p); };
A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); };  \\ From A046523
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; };
v304744 = rgs_transform(vector(65538,n,A046523(A052330(n-1))));
A304744(n) = v304744[1+n];

A318832 Restricted growth sequence transform of A278222(A000203(n)).

Original entry on oeis.org

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

Views

Author

Antti Karttunen, Sep 04 2018

Keywords

Comments

Sequence allots a distinct value for each distinct multiset formed from the lengths of 1-runs in the binary representation of A000203(n).
For all i, j: a(i) = a(j) => A175548(i) = A175548(j).

Links

Crossrefs

Cf. A000203, A278222.
Cf. also A286622, A304101, A318831.

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));
v318832 = rgs_transform(vector(up_to,n,A278222(sigma(n))));
A318832(n) = v318832[n];

A337200 Lexicographically earliest infinite sequence such that a(i) = a(j) => A278222(A337194(i)) = A278222(A337194(j)), for all i, j >= 1.

Original entry on oeis.org

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

Views

Author

Antti Karttunen, Aug 20 2020

Keywords

Comments

Restricted growth sequence transform of f(n) = A278222(A337194(n)).
For all i, j: A324400(i) = A324400(j) => a(i) = a(j) => A337199(i) = A337199(j).

Links

Crossrefs

Cf. A000203, A000265, A278222, A161942, A336698, A337194, A337199.
Cf. also A286622, A324400, A337201.

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));
A337194(n) = (1+A000265(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));
v337200 = rgs_transform(vector(up_to, n, A278222(A337194(n))));
A337200(n) = v337200[n];
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