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 1-6 of 6 results.

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");

A300235 Restricted growth sequence transform of A291765, filter combining A001065(n) and A046523(n), the sum of proper divisors and the prime signature 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, 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, 47, 56, 2, 57, 58, 59, 2, 60, 41, 61, 62, 63, 2, 64, 37, 65, 66, 67, 68, 69, 2, 70, 71
Offset: 1

Views

Author

Antti Karttunen, Mar 01 2018

Keywords

Examples

			a(51) = a(91) (= 33) because both are nonsquare semiprimes (3*17 and 7*13), and the sum of their proper divisors (A001065) are equal 1+3+17 = 1+7+13 = 21.

Links

Crossrefs

Cf. A001065, A046523, A291765.
Cf. also A295885, A300223, A300226, A300229, A300230, A300231, A300232, A300233.

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])); }
A001065(n) = (sigma(n)-n);
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
A291765(n) = (1/2)*(2 + ((A001065(n)+A046523(n))^2) - A001065(n) - 3*A046523(n));
write_to_bfile(1,rgs_transform(vector(up_to,n,A291765(n))),"b300235.txt");

A300231 Filter sequence combining A001065(n) and A009194(n), the sum of proper divisors of n and gcd(n,sigma(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, 19, 25, 2, 13, 26, 27, 2, 28, 2, 29, 30, 31, 2, 32, 33, 34, 12, 35, 2, 36, 26, 37, 38, 39, 2, 40, 2, 41, 42, 43, 44, 45, 2, 46, 47, 48, 2, 49, 2, 50, 51, 52, 44, 53, 2, 54, 55, 56, 2, 57, 38, 35, 30, 58, 2, 59, 60, 32, 61, 62, 63, 64
Offset: 1

Views

Author

Antti Karttunen, Mar 01 2018

Keywords

Comments

Restricted growth sequence transform of P(A001065(n), A009194(n)), where P(a,b) is a two-argument form of A000027 used as a Cantor pairing function N x N -> N.

Examples

			a(27) = a(35) (= 19) because A001065(27) = A001065(35) = 13 and A009194(27) = A009194(35) = 1.

Links

Crossrefs

Cf. A001065, A009194.
Cf. also A295885, A300223, A300226, A300229, A300230, A300232, A300233, A300235.

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])); }
A001065(n) = (sigma(n)-n);
A009194(n) = gcd(n, sigma(n));
Aux300231(n) = (1/2)*(2 + ((A001065(n)+A009194(n))^2) - A001065(n) - 3*A009194(n));
write_to_bfile(1,rgs_transform(vector(65537,n,Aux300231(n))),"b300231.txt");

A300232 Restricted growth sequence transform of A286152, filter combining A051953(n) and A046523(n), cototient and the prime signature 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, 12, 13, 14, 2, 15, 16, 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, 36, 37, 2, 38, 27, 39, 40, 41, 2, 42, 2, 43, 44, 45, 46, 47, 2, 48, 49, 47, 2, 50, 2, 51, 52, 53, 46, 54, 2, 55, 56, 57, 2, 58, 40, 59, 60, 61, 2, 62, 36, 63, 64, 65, 66, 67, 2, 68, 69
Offset: 1

Views

Author

Antti Karttunen, Mar 01 2018

Keywords

Examples

			a(39) = a(55) (= 27) because both are nonsquare semiprimes (3*13 and 5*11), and both have cototient value 15 = 39 - phi(39) = 55 - phi(55).

Links

Crossrefs

Cf. A046523, A051953, A286152.
Cf. also A295885, A300223, A300224, A300226, A300229, A300230, A300231, A300233, A300235.

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])); }
A051953(n) = (n - eulerphi(n));
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
A286152(n) = (2 + ((A051953(n)+A046523(n))^2) - A051953(n) - 3*A046523(n))/2;
write_to_bfile(1,rgs_transform(vector(up_to,n,A286152(n))),"b300232.txt");

A300233 Filter sequence combining A051953(n) and A009194(n), cototient of n and gcd(n,sigma(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, 13, 2, 15, 16, 17, 14, 18, 2, 19, 2, 20, 21, 22, 23, 24, 2, 25, 26, 27, 2, 28, 2, 29, 30, 31, 2, 32, 33, 34, 35, 36, 2, 37, 26, 38, 39, 40, 2, 41, 2, 42, 43, 44, 45, 46, 2, 47, 48, 49, 2, 50, 2, 51, 52, 53, 45, 54, 2, 55, 43, 56, 2, 57, 39, 58, 59, 60, 2, 61, 62, 60, 63, 55, 64, 65, 2, 66
Offset: 1

Views

Author

Antti Karttunen, Mar 01 2018

Keywords

Comments

Restricted growth sequence transform of P(A051953(n), A009194(n)), where P(a,b) is a two-argument form of A000027 used as a Cantor pairing function N x N -> N.

Examples

			a(20) = a(22) (= 13) because A051953(20) = A051953(22) = 12 and A009194(20) = A009194(22) = 2.

Links

Crossrefs

Cf. A009194, A051953.
Cf. also A295885, A300223, A300226, A300229, A300230, A300231, A300232, A300235.

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])); }
A009194(n) = gcd(n, sigma(n));
A051953(n) = (n - eulerphi(n));
Aux300233(n) = (1/2)*(2 + ((A051953(n)+A009194(n))^2) - A051953(n) - 3*A009194(n));
write_to_bfile(1,rgs_transform(vector(65537,n,Aux300233(n))),"b300233.txt");

A305895 Filter sequence combining sum of proper divisors (A001065) and cototient (A051953) 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, 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, 47, 56, 2, 57, 58, 59, 2, 60, 41, 61, 62, 63, 2, 64, 37, 65, 66, 67, 68, 69, 2, 70, 71, 72, 2, 73, 2, 74, 75
Offset: 1

Views

Author

Antti Karttunen, Jun 14 2018

Keywords

Comments

Restricted growth sequence transform of ordered pair [A001065(n), A051953(n)].

Links

Crossrefs

Cf. A001065, A051953, A295885, A305800.
Differs from A300249 for the first time at n=105, where a(105) = 75, while A300249(105) = 56.
Differs from A300235 for the first time at n=153, where a(153) = 110, while A300235(153) = 106.

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; };
A001065(n) = (sigma(n)-n);
A051953(n) = (n-eulerphi(n));
Aux305895(n) = [A001065(n), A051953(n)];
v305895 = rgs_transform(vector(up_to,n,Aux305895(n)));
A305895(n) = v305895[n];

Formula

a(1) = 1; for n > 1, a(n) = 1 + A295885(n).
Showing 1-6 of 6 results.

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