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 1-9 of 9 results.

A293225 Compound filter: a(n) = P(A293224(n), A293223(n)), where P(n,k) is sequence A000027 used as a pairing function.

Original entry on oeis.org

1, 2, 2, 5, 2, 8, 2, 12, 4, 13, 2, 32, 2, 40, 30, 33, 2, 59, 2, 58, 42, 69, 2, 143, 8, 80, 29, 83, 2, 178, 2, 197, 38, 96, 25, 239, 2, 100, 121, 163, 2, 221, 2, 202, 194, 103, 2, 448, 61, 365, 59, 245, 2, 333, 48, 576, 187, 256, 2, 720, 2, 278, 546, 718, 138, 606, 2, 503, 114, 1009, 2, 1101, 2, 437, 651, 678, 532, 831, 2, 1400, 172, 213, 2, 1508, 71, 500, 597
Offset: 1

Views

Author

Antti Karttunen, Oct 03 2017

Keywords

Links

Crossrefs

Cf. A000027, A019565, A293221, A293222, A293223, A293224, A293226 (rgs-version of this filter).
Cf. also A289813, A289814, A290093, A290094.

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; };
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; };
v293223 = rgs_transform(vector(19683,n,A293221(n)));
A293223(n) = v293223[n];
v293224 = rgs_transform(vector(19683,n,A293222(n)));
A293224(n) = v293224[n];
A293225(n) = (1/2)*(2 + ((A293224(n) + A293223(n))^2) - A293224(n) - 3*A293223(n));
  • Scheme
    (define (A293225 n) (* 1/2 (+ (expt (+ (A293224 n) (A293223 n)) 2) (- (A293224 n)) (- (* 3 (A293223 n))) 2)))

Formula

a(n) = (1/2)*(2 + ((A293224(n) + A293223(n))^2) - A293224(n) - 3*A293223(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");

A293215 Restricted growth sequence transform of A293214, where A293214(n) = Product_{d|n, dA019565(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, 28, 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, 73
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, A019565, A048675, A293214, A293217 (a variant), A293223, A293224, A293226.
Differs from related A293232 for the first time at n=55, where a(55) = 28, while A293232(55) = 39.

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
A293214(n) = { my(m=1); fordiv(n,d,if(d < n,m *= A019565(d))); m; };
write_to_bfile(1,rgs_transform(vector(16384,n,A293214(n))),"b293215.txt");

A293221 a(n) = Product_{d|n, dA019565(A289813(d)); a product obtained from the 1-digits present in ternary expansions of proper divisors of n.

Original entry on oeis.org

1, 2, 2, 2, 2, 6, 2, 12, 6, 6, 2, 36, 2, 4, 18, 12, 2, 30, 2, 360, 12, 10, 2, 540, 6, 60, 30, 360, 2, 900, 2, 120, 30, 10, 12, 2700, 2, 4, 180, 360, 2, 540, 2, 360, 450, 6, 2, 5400, 4, 120, 30, 360, 2, 210, 30, 5040, 12, 14, 2, 1701000, 2, 84, 180, 2520, 180, 1260, 2, 840, 18, 12600, 2, 94500, 2, 140, 180, 840, 20, 18900, 2, 756000, 210, 210, 2, 23814000, 30
Offset: 1

Views

Author

Antti Karttunen, Oct 03 2017

Keywords

Links

Crossrefs

Cf. A019565, A289813, A293214, A293222, A293223 (restricted growth sequence transform), A293226.
Cf. also A290091.

Programs

  • PARI
    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; };

Formula

a(n) = Product_{d|n, dA019565(A289813(d)).
For all n >= 0, a(3^n) = A002110(n).

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");

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");

A320011 Filter sequence combined from those proper divisors d of n for which +1 == d (mod 3); Restricted growth sequence transform of A319991.

Original entry on oeis.org

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

Views

Author

Antti Karttunen, Oct 03 2018

Keywords

Comments

For all i, j:
a(i) = a(j) => A320001(i) = A320001(j),
a(i) = a(j) => A293897(i) = A293897(j).

Links

Crossrefs

Cf. A293223, A319991, A320010, A320012, A320013, A320014.

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(j,v); factorback(Mat(vector(if(n, #n=vecextract(binary(n), "-1..1")), j, [prime(j), n[j]])~))}; \\ From A019565
A319991(n) = { my(m=1); fordiv(n,d,if((dA019565(d))); m; };
v320011 = rgs_transform(vector(up_to,n,A319991(n)));
A320011(n) = v320011[n];

A318835 Restricted growth sequence transform of A318834, product_{d|n, dA019565(A000010(d)).

Original entry on oeis.org

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

Views

Author

Antti Karttunen, Sep 04 2018

Keywords

Comments

For all i, j: a(i) = a(j) => A051953(i) = A051953(j).

Links

Crossrefs

Cf. A000010, A019565, A318831, A318834,
Cf. also A293215, A293217, A293223, A293224, A293232, A300833, A300835.

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(j,v); factorback(Mat(vector(if(n, #n=vecextract(binary(n), "-1..1")), j, [prime(j), n[j]])~))}; \\ From A019565
A318834(n) = { my(m=1); fordiv(n,d,if(d < n,m *= A019565(eulerphi(d)))); m; };
v318835 = rgs_transform(vector(up_to,n,A318834(n)));
A318835(n) = v318835[n];

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];
Showing 1-9 of 9 results.

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