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 1-10 of 14 results. Next

A293214 a(n) = Product_{d|n, dA019565(d).

Original entry on oeis.org

1, 2, 2, 6, 2, 36, 2, 30, 12, 60, 2, 2700, 2, 180, 120, 210, 2, 7560, 2, 6300, 360, 252, 2, 661500, 20, 420, 168, 94500, 2, 23814000, 2, 2310, 504, 132, 600, 43659000, 2, 396, 840, 2425500, 2, 187110000, 2, 207900, 352800, 1980, 2, 560290500, 60, 194040, 264, 485100, 2, 115259760, 840, 254677500, 792, 4620, 2, 264737261250000, 2, 13860
Offset: 1

Views

Author

Antti Karttunen, Oct 03 2017

Keywords

Links

Crossrefs

Cf. A001065, A002110, A019565, A048675, A091954, A292257, A293215 (restricted growth sequence transform).
Cf. also A293216, A293221, A293222, A293225, A293231, A293442, A300830, A300831, A300832, A300833, A300834.

Programs

  • PARI
    A019565(n) = {my(j,v); factorback(Mat(vector(if(n, #n=vecextract(binary(n), "-1..1")), j, [prime(j), n[j]])~))}; \\ From A019565
A293214(n) = { my(m=1); fordiv(n,d,if(d < n,m *= A019565(d))); m; };

Formula

a(n) = Product_{d|n, dA019565(d).
a(n) = A300830(n) * A300831(n) * A300832(n). - Antti Karttunen, Mar 16 2018
Other identities.
For n >= 0, a(2^n) = A002110(n).
For n >= 1:
A048675(a(n)) = A001065(n).
A001222(a(n)) = A292257(n).
A007814(a(n)) = A091954(n).
A087207(a(n)) = A218403(n).
A248663(a(n)) = A227320(n).

A293226 Restricted growth sequence transform of A293225, a filter combining two products, the other formed from the 1-digits (A293221) and the other from the 2-digits (A293222) present in the ternary expansions of proper divisors of n.

Original entry on oeis.org

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

Views

Author

Antti Karttunen, Oct 03 2017

Keywords

Comments

For all i, j: a(i) = a(j) => A001065(i) = A001065(j).

Links

Crossrefs

Cf. A001065, A293221, A293222, A293223, A293224, A293225.
Cf. also A290093, A290094, A293215, A293217, A293232.

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; };
write_to_bfile(start_offset,vec,bfilename) = { for(n=1, length(vec), write(bfilename, (n+start_offset)-1, " ", vec[n])); }
A019565(n) = {my(j,v); factorback(Mat(vector(if(n, #n=vecextract(binary(n), "-1..1")), j, [prime(j), n[j]])~))}; \\ This function from M. F. Hasler
A289813(n) = { my (d=digits(n, 3)); fromdigits(vector(#d, i, if (d[i]==1, 1, 0)), 2); };
A289814(n) = { my (d=digits(n, 3)); fromdigits(vector(#d, i, if (d[i]==2, 1, 0)), 2); };
A293221(n) = { my(m=1); fordiv(n,d,if(d < n,m *= A019565(A289813(d)))); m; };
A293222(n) = { my(m=1); fordiv(n,d,if(d < n,m *= A019565(A289814(d)))); m; };
Anot_submitted(n) = (1/2)*(2 + ((A293222(n) + A293221(n))^2) - A293222(n) - 3*A293221(n)); \\ Eq.class-wise equal to A293225.
write_to_bfile(1,rgs_transform(vector(19683,n,Anot_submitted(n))),"b293226.txt");

A300833 Filter sequence combining A300830(n), A300831(n) and A300832(n), three products formed from such proper divisors d of n for which mu(n/d) = 0, +1 or -1 respectively, where mu is Möbius mu function (A008683).

Original entry on oeis.org

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

Views

Author

Antti Karttunen, Mar 16 2018

Keywords

Comments

Restricted growth sequence transform of triple [A300830(n), A300831(n), A300832(n)].
For all i, j:
a(i) = a(j) => A293215(i) = A293215(j) => A001065(i) = A001065(j).
a(i) = a(j) => A051953(i) = A051953(j).
a(i) = a(j) => A295885(i) = A295885(j).
Apparently this is also the restricted growth sequence transform of ordered pair [A300831(n), A300832(n)], which is true if it holds that whenever we have A300831(i) = A300831(j) and A300832(i) = A300832(j) for any i, j, then also A300830(i) = A300830(j). This has been checked for the first 65537 terms.

Examples

			a(39) = a(55) (= 28) as A300830(39) = A300830(55) = 1, A300831(39) = A300831(55) = 2 and A300832(39) = A300832(55) = 420.

Links

Crossrefs

Cf. A019565, A051953, A300831, A300832.
Cf. also A293214, A293215, A293226, A295885, A300825.

Programs

  • PARI
    allocatemem(2^30);
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; };
write_to_bfile(start_offset,vec,bfilename) = { for(n=1, length(vec), write(bfilename, (n+start_offset)-1, " ", vec[n])); }
A019565(n) = {my(j,v); factorback(Mat(vector(if(n, #n=vecextract(binary(n), "-1..1")), j, [prime(j), n[j]])~))}; \\ From A019565
A300830(n) = { my(m=1); fordiv(n,d,if(!moebius(n/d),m *= A019565(d))); m; };
A300831(n) = { my(m=1); fordiv(n,d,if((d < n)&&(1==moebius(n/d)), m *= A019565(d))); m; };
A300832(n) = { my(m=1); fordiv(n,d,if(-1==moebius(n/d), m *= A019565(d))); m; };
Aux300833(n) = [A300830(n), A300831(n), A300832(n)];
write_to_bfile(1,rgs_transform(vector(up_to,n,Aux300833(n))),"b300833.txt");

A293217 Restricted growth sequence transform of A293216, where A293216(n) = Product_{d|n, dA260443(d).

Original entry on oeis.org

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

Views

Author

Antti Karttunen, Oct 03 2017

Keywords

Comments

For all i, j: a(i) = a(j) => A001065(i) = A001065(j).

Links

Crossrefs

Cf. A001065, A260443, A293216, A293215 (a variant).

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; };
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
A260443(n) = if(n<2, n+1, if(n%2, A260443(n\2)*A260443(n\2+1), A003961(A260443(n\2))));
A293216(n) = { my(m=1); fordiv(n,d,if(d < n,m *= A260443(d))); m; };
write_to_bfile(1,rgs_transform(vector(16384,n,A293216(n))),"b293217.txt");

A293223 Restricted growth sequence transform of A293221, a product formed from the 1-digits present in the ternary expansion of proper divisors of n.

Original entry on oeis.org

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

Views

Author

Antti Karttunen, Oct 03 2017

Keywords

Links

Crossrefs

Cf. A293221.
Cf. also A290091, A293224, A293225, A293226, A293215, A293217, A293232.

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; };
write_to_bfile(start_offset,vec,bfilename) = { for(n=1, length(vec), write(bfilename, (n+start_offset)-1, " ", vec[n])); }
A019565(n) = {my(j,v); factorback(Mat(vector(if(n, #n=vecextract(binary(n), "-1..1")), j, [prime(j), n[j]])~))}; \\ This function from M. F. Hasler
A289813(n) = { my (d=digits(n, 3)); fromdigits(vector(#d, i, if (d[i]==1, 1, 0)), 2); } \\ From Remy Sigrist
A293221(n) = { my(m=1); fordiv(n,d,if(d < n,m *= A019565(A289813(d)))); m; };
write_to_bfile(1,rgs_transform(vector(19683,n,A293221(n))),"b293223.txt");

A293224 Restricted growth sequence transform of A293222, a product formed from the 2-digits present in the ternary expansions of proper divisors of n.

Original entry on oeis.org

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

Views

Author

Antti Karttunen, Oct 03 2017

Keywords

Links

Crossrefs

Cf. A293222.
Cf. also A290092, A293223, A293215, A293217, A293232, A293225, A293226.

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; };
write_to_bfile(start_offset,vec,bfilename) = { for(n=1, length(vec), write(bfilename, (n+start_offset)-1, " ", vec[n])); }
A019565(n) = {my(j,v); factorback(Mat(vector(if(n, #n=vecextract(binary(n), "-1..1")), j, [prime(j), n[j]])~))}; \\ This function from M. F. Hasler
A289814(n) = { my (d=digits(n, 3)); fromdigits(vector(#d, i, if (d[i]==2, 1, 0)), 2); } \\ From Remy Sigrist
A293222(n) = { my(m=1); fordiv(n,d,if(d < n,m *= A019565(A289814(d)))); m; };
write_to_bfile(1,rgs_transform(vector(19683,n,A293222(n))),"b293224.txt");

A305793 Restricted growth sequence transform of A305792, a filter sequence constructed from binary expansions of the proper divisors of n.

Original entry on oeis.org

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

Views

Author

Antti Karttunen, Jun 11 2018

Keywords

Links

Crossrefs

Cf. A278222, A286622, A305792, A305795.
Cf. also A293215, A300833, A304103, A305813.

Programs

  • PARI
    \\ Needs also code from A286622:
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; };
A305792(n) = { my(m=1); fordiv(n,d,if(dA286622(d)-1))); (m); };
v305793 = rgs_transform(vector(up_to, n, A305792(n)));
A305793(n) = v305793[n];

Formula

For all i, j:
a(i) = a(j) => A000005(i) = A000005(j).
a(i) = a(j) => A292257(i) = A292257(j).
a(i) = a(j) => A305426(i) = A305426(j).
a(i) = a(j) => A305435(i) = A305435(j).

A293232 Restricted growth sequence transform of A293231, where A293231(n) = Product_{d|n, dA019565(A193231(d)).

Original entry on oeis.org

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

Views

Author

Antti Karttunen, Oct 03 2017

Keywords

Links

Crossrefs

Cf. A019565, A193231.
Cf. A290090.
Differs from related A293215 for the first time at n=55, where a(55) = 39, while A293215(55) = 28.

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; };
write_to_bfile(start_offset,vec,bfilename) = { for(n=1, length(vec), write(bfilename, (n+start_offset)-1, " ", vec[n])); }
A019565(n) = {my(j,v); factorback(Mat(vector(if(n, #n=vecextract(binary(n), "-1..1")), j, [prime(j), n[j]])~))}; \\ This function from M. F. Hasler
A193231(n) = { my(x='x); subst(lift(Mod(1, 2)*subst(Pol(binary(n), x), x, 1+x)), x, 2) }; \\ This function from Franklin T. Adams-Watters
A293231(n) = { my(m=1); fordiv(n,d,if(d < n,m *= A019565(A193231(d)))); m; };
write_to_bfile(1,rgs_transform(vector(65537,n,A293231(n))),"b293232.txt");

A227320 Binary XOR of proper divisors of n.

Original entry on oeis.org

0, 1, 1, 3, 1, 0, 1, 7, 2, 6, 1, 2, 1, 4, 7, 15, 1, 15, 1, 8, 5, 8, 1, 6, 4, 14, 11, 14, 1, 6, 1, 31, 9, 18, 3, 21, 1, 16, 15, 20, 1, 26, 1, 26, 1, 20, 1, 14, 6, 21, 19, 16, 1, 6, 15, 26, 17, 30, 1, 4, 1, 28, 25, 63, 9, 58, 1, 52, 21, 38, 1, 33, 1, 38, 17, 50, 13, 54, 1
Offset: 1

Views

Author

Alex Ratushnyak, Jul 06 2013

Keywords

Comments

An alternative definition (with A027751) would define a(1)=1. - R. J. Mathar, Jul 14 2013
However, this definition is more aligned with A001065 and A218403 where the initial term a(1) is also 0. - Antti Karttunen, Oct 08 2017

Links

Crossrefs

Cf. A001065, A027751, A178910, A218403, A248663, A293214, A293215.

Programs

  • Mathematica
    Array[BitXor @@ Most@ Divisors@ # &, 79] (* Michael De Vlieger, Oct 08 2017 *)
  • PARI
    A227320(n) = { my(s=0); fordiv(n,d,if(dAntti Karttunen, Oct 08 2017

Formula

a(n) = A178910(n) XOR n, where XOR is the binary logical exclusive or operator.
From Antti Karttunen, Oct 08 2017: (Start)
a(n) = A248663(A293214(n)).
a(n) <= A218403(n) <= A001065(n).
(End)

A300835 Restricted growth sequence transform of A300834, product_{d|n, dA019565(A003714(d)); Filter sequence related to Zeckendorf-representations of proper divisors of n.

Original entry on oeis.org

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

Views

Author

Antti Karttunen, Mar 18 2018

Keywords

Comments

For all i, j: a(i) = a(j) => A001065(i) = A001065(j).
For all i, j: a(i) = a(j) => A300836(i) = A300836(j).

Examples

			For cases n=10 and 49, we see that 10 has proper divisors 1, 2 and 5 and these have Zeckendorf-representations (A014417) 1, 10 and 1000, while 49 has proper divisors 1 and 7 and these have Zeckendorf-representations 1 and 1010. When these Zeckendorf-representations are summed (columnwise without carries), result in both cases is 1011, thus a(10) = a(49).

Links

Crossrefs

Cf. A003714, A014417, A019565, A300834, A300836.
Cf. also A293215, A293217, A293223, A293224, A293232, A300833 for similar filtering sequences.

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; };
write_to_bfile(start_offset,vec,bfilename) = { for(n=1, length(vec), write(bfilename, (n+start_offset)-1, " ", vec[n])); }
A072649(n) = { my(m); if(n<1, 0, m=0; until(fibonacci(m)>n, m++); m-2); }; \\ From A072649
A003714(n) = { my(s=0,w); while(n>2, w = A072649(n); s += 2^(w-1); n -= fibonacci(w+1)); (s+n); }
A019565(n) = {my(j,v); factorback(Mat(vector(if(n, #n=vecextract(binary(n), "-1..1")), j, [prime(j), n[j]])~))}; \\ From A019565
A300834(n) = { my(m=1); fordiv(n,d,if(d < n,m *= A019565(A003714(d)))); m; };
write_to_bfile(1,rgs_transform(vector(up_to,n,A300834(n))),"b300835.txt");
Showing 1-10 of 14 results. Next

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