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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A305303 Restricted growth sequence transform of ordered pair [A278222(A304760(n)), A278222(A291760(n))], constructed from runlengths of 1-digits and 2-digits in base-3 representation of A254103(n).

Original entry on oeis.org

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

Views

Author

Antti Karttunen, May 30 2018

Keywords

Comments

Restricted growth sequence transform of A290093(A254103(n)).
For all i, j: a(i) = a(j) => A286633(i) = A286633(j) => A286632(i) = A286632(j).

Links

Crossrefs

Cf. A254103, A289813, A289814, A291760, A304740, A304746, A304760.
Cf. also A286632, A286633, A290093, A290094, A305301, A305302.

Programs

  • PARI
    A254103(n) = if(!n,n,if(!(n%2),(3*A254103(n/2))-1,(3*(1+A254103((n-1)/2)))\2));
A289813(n) = { my (d=digits(n, 3)); fromdigits(vector(#d, i, if (d[i]==1, 1, 0)), 2); } \\ From A289813
A289814(n) = { my (d=digits(n, 3)); fromdigits(vector(#d, i, if (d[i]==2, 1, 0)), 2); } \\ From A289813
A304760(n) = A289813(A254103(n));
A291760(n) = A289814(A254103(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));
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; };
Aux305303(n) = [A278222(A304760(n)), A278222(A291760(n))];
v305303 = rgs_transform(vector(65538,n,Aux305303(n-1)));
A305303(n) = v305303[1+n];

A305433 Restricted growth sequence transform of ordered pair [A278222(A305295(n)), A278222(A291763(n))], constructed from runlengths of 1-digits and 2-digits in base-3 representation of A245612(n).

Original entry on oeis.org

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

Views

Author

Antti Karttunen, Jun 01 2018

Keywords

Comments

Restricted growth sequence transform of A290093(A245612(n)).

Links

Crossrefs

Cf. A245612, A278222, A290093, A290094, A291763, A305295, A305431, A305432, A305434.
Cf. also A305303.

Programs

  • PARI
    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;
A254049(n) = A048673((2*n)-1);
A245612(n) = if(n<2,1+n,if(!(n%2),(3*A245612(n/2))-1,A254049(A245612((n-1)/2))));
A289813(n) = { my (d=digits(n, 3)); fromdigits(vector(#d, i, if (d[i]==1, 1, 0)), 2); } \\ From A289813
A289814(n) = { my (d=digits(n, 3)); fromdigits(vector(#d, i, if (d[i]==2, 1, 0)), 2); } \\ From A289813
A305295(n) = A289813(A245612(n));
A291763(n) = A289814(A245612(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));
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; };
Aux305433(n) = [A278222(A305295(n)), A278222(A291763(n))];
v305433 = rgs_transform(vector(65538,n,Aux305433(n-1)));
A305433(n) = v305433[1+n];

A340381 Lexicographically earliest infinite sequence such that a(i) = a(j) => A278222(A304759(i)) = A278222(A304759(j)), for all i, j >= 1.

Original entry on oeis.org

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

Views

Author

Antti Karttunen, Jan 16 2021

Keywords

Comments

For all i, j: A304758(i) = A304758(j) => a(i) = a(j) => A340378(i) = A340378(j).

Links

Crossrefs

Cf. A278222, A304758, A304759, A340378, A340382, A340383.
Cf. A340376 (positions of 2's).
Cf. also A305301.

Programs

  • PARI
    up_to = 65537;
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;
A289813(n) = { my (d=digits(n, 3)); fromdigits(vector(#d, i, if (d[i]==1, 1, 0)), 2); } \\ From A289813
A304759(n) = A289813(A048673(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));
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; };
v340381 = rgs_transform(vector(up_to,n,A278222(A304759(n))));
A340381(n) = v340381[n];

A304758 Restricted growth sequence transform of A304759(n), formed from 1-digits in ternary representation of A048673(n).

Original entry on oeis.org

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

Views

Author

Antti Karttunen, May 30 2018

Keywords

Links

Crossrefs

Cf. A048673, A289813, A304759.
Cf. also A304740, A304748.

Programs

  • PARI
    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;
A289813(n) = { my (d=digits(n, 3)); from digits(vector(#d, i, if (d[i]==1, 1, 0)), 2); } \\ From A289813
A304759(n) = A289813(A048673(n));
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; };
v304758 = rgs_transform(vector(65537,n,A304759(n)));
A304758(n) = v304758[n];

A305431 Restricted growth sequence transform of A278222(A305295(n)), constructed from runlengths of 1-digits in base-3 representation of A245612(n).

Original entry on oeis.org

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

Views

Author

Antti Karttunen, Jun 01 2018

Keywords

Links

Crossrefs

Cf. A245612, A278222, A305295, A305296, A305432, A305433.
Cf. also A305301.

Programs

  • PARI
    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;
A254049(n) = A048673((2*n)-1);
A245612(n) = if(n<2,1+n,if(!(n%2),(3*A245612(n/2))-1,A254049(A245612((n-1)/2))));
A289813(n) = { my (d=digits(n, 3)); fromdigits(vector(#d, i, if (d[i]==1, 1, 0)), 2); } \\ From A289813
A305295(n) = A289813(A245612(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));
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; };
v305431 = rgs_transform(vector(65538,n,A278222(A305295(n-1))));
A305431(n) = v305431[1+n];

A351031 a(n) = Product_{d|n, dA019565(A304759(d)).

Original entry on oeis.org

1, 2, 2, 2, 2, 6, 2, 6, 6, 12, 2, 18, 2, 2, 36, 90, 2, 180, 2, 180, 6, 4, 2, 810, 12, 10, 180, 30, 2, 180, 2, 9450, 12, 20, 12, 56700, 2, 30, 30, 56700, 2, 420, 2, 12, 1080, 10, 2, 1275750, 2, 120, 60, 30, 2, 31500, 24, 9450, 90, 20, 2, 238140, 2, 4, 2520, 10914750, 60, 84, 2, 420, 30, 31500, 2, 2946982500, 2, 6
Offset: 1

Views

Author

Antti Karttunen, Jan 29 2022

Keywords

Links

Crossrefs

Cf. A019565, A048673, A289813, A304759, A351030, A351032, A351033 (rgs-transform).
Cf. also A293221.

Programs

  • PARI
    A019565(n) = { my(m=1, p=1); while(n>0, p = nextprime(1+p); if(n%2, m *= p); n >>= 1); (m); };
A048673(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); (1+factorback(f))/2; };
A289813(n) = { my(d=digits(n, 3)); fromdigits(vector(#d, i, if (d[i]==1, 1, 0)), 2); } \\ From A289813
A304759(n) = A289813(A048673(n));
A351031(n) = { my(m=1); fordiv(n,d,if(dA019565(A304759(d)))); (m); };

A351033 Lexicographically earliest infinite sequence such that a(i) = a(j) => A351031(i) = A351031(j), for all i, j >= 1.

Original entry on oeis.org

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

Views

Author

Antti Karttunen, Jan 29 2022

Keywords

Comments

Restricted growth sequence transform of A351031.

Links

Crossrefs

Cf. A019565, A048673, A289813, A304759, A351030, A351031, A351034.
Cf. also A293223.

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; };
A019565(n) = { my(m=1, p=1); while(n>0, p = nextprime(1+p); if(n%2, m *= p); n >>= 1); (m); };
A048673(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); (1+factorback(f))/2; };
A289813(n) = { my(d=digits(n, 3)); fromdigits(vector(#d, i, if (d[i]==1, 1, 0)), 2); } \\ From A289813
A304759(n) = A289813(A048673(n));
A351031(n) = { my(m=1); fordiv(n,d,if(dA019565(A304759(d)))); (m); };
v351033 = rgs_transform(vector(up_to, n, A351031(n)));
A351033(n) = v351033[n];

A351093 Lexicographically earliest infinite sequence such that a(i) = a(j) => A351091(i) = A351091(j), for all i, j >= 1.

Original entry on oeis.org

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

Views

Author

Antti Karttunen, Jan 31 2022

Keywords

Comments

Restricted growth sequence transform of A351091.

Links

Crossrefs

Cf. A351090, A351091, A351094.

Programs

  • PARI
    up_to = 20000;
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; };
A019565(n) = { my(m=1, p=1); while(n>0, p = nextprime(1+p); if(n%2, m *= p); n >>= 1); (m); };
A289813(n) = { my(d=digits(n, 3)); fromdigits(vector(#d, i, if (d[i]==1, 1, 0)), 2); }; \\ From A289813
A351091(n) = { my(m=1); fordiv(n>>valuation(n,2),d,m *= A019565(A289813(d))); (m); };
v351093 = rgs_transform(vector(up_to, n, A351091(n)));
A351093(n) = v351093[n];

A332412 a(n) is the real part of f(n) = Sum_{d_k > 0} 3^k * i^(d_k-1) where Sum_{k >= 0} 5^k * d_k is the base 5 representation of n and i denotes the imaginary unit. Sequence A332413 gives imaginary parts.

Original entry on oeis.org

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

Views

Author

Rémy Sigrist, Feb 12 2020

Keywords

Comments

The representation of {f(n)} corresponds to the cross form of the Vicsek fractal.
As a set, {f(n)} corresponds to the Gaussian integers whose real and imaginary parts have not simultaneously a nonzero digit at the same place in their balanced ternary representations.

Examples

			For n = 103:
- 103 = 4*5^2 + 3*5^0,
- so f(123) = 3^2 * i^(4-1) + 3^0 * i^(3-1) = -1 - 9*i,
- and a(n) = -1.

Links

Crossrefs

See A332497 for a similar sequence.
Cf. A031219, A289813, A332413 (imaginary parts).

Programs

  • PARI
    a(n) = { my (d=Vecrev(digits(n,5))); real(sum (k=1, #d, if (d[k], 3^(k-1)*I^(d[k]-1), 0))) }

Formula

a(n) = 0 iff the n-th row of A031219 has only even terms.
a(5*n) = 3*a(n).
a(5*n+1) = 3*a(n) + 1.
a(5*n+2) = 3*a(n).
a(5*n+3) = 3*a(n) - 1.
a(5*n+4) = 3*a(n).

A340383 Lexicographically earliest infinite sequence such that a(i) = a(j) => f(i) = f(j), where f(n) = [A278222(A304759(n)), A278222(A291759(n))], for all i, j >= 1.

Original entry on oeis.org

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

Views

Author

Antti Karttunen, Jan 16 2021

Keywords

Comments

Restricted growth sequence transform of the ordered pair [A340381(n), A340382(n)], or equally, of the function f(n) = A290093(A048673(n)).
For all i, j: a(i) = a(j) => A286586(i) = A286586(j) => A286585(i) = A286585(j).

Links

Crossrefs

Cf. A048673, A278222, A290093, A291759, A304759, A340381, A340382, A340684.
Cf. also A290094, A305303.

Programs

  • PARI
    up_to = 65537;
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;
A289814(n) = { my (d=digits(n, 3)); fromdigits(vector(#d, i, if (d[i]==2, 1, 0)), 2); } \\ From A289814
A291759(n) = A289814(A048673(n));
A289813(n) = { my (d=digits(n, 3)); fromdigits(vector(#d, i, if (d[i]==1, 1, 0)), 2); } \\ From A289813
A304759(n) = A289813(A048673(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));
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; };
Aux340383(n) = [A278222(A291759(n)),A278222(A304759(n))];
v340383 = rgs_transform(vector(up_to,n,Aux340383(n)));
A340383(n) = v340383[n];
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