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.

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");

A300229 Restricted growth sequence transform of A285729, combining A032742(n) and A046523(n), the largest proper divisor 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, 7, 10, 2, 11, 2, 12, 9, 13, 2, 14, 15, 16, 17, 18, 2, 19, 2, 20, 13, 21, 9, 22, 2, 23, 16, 24, 2, 25, 2, 26, 27, 28, 2, 29, 30, 31, 21, 32, 2, 33, 13, 34, 23, 35, 2, 36, 2, 37, 38, 39, 16, 40, 2, 41, 28, 42, 2, 43, 2, 44, 31, 45, 13, 46, 2, 47, 48, 49, 2, 50, 21, 51, 35, 52, 2, 53, 16, 54, 37, 55, 23, 56, 2, 57
Offset: 1

Views

Author

Antti Karttunen, Mar 01 2018

Keywords

Examples

			a(10) = a(15) (= 7) because both are nonsquare semiprimes (2*5 and 3*5), and when the smallest prime factor is divided out, both yield the same quotient, 5.

Links

Crossrefs

Cf. A032742, A046523, A285729.
Cf. also A300223, A300226, A300230, A300231, 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])); }
A032742(n) = if(1==n,n,n/vecmin(factor(n)[,1]));
A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); };  \\ From A046523
A285729(n) = (1/2)*(2 + ((A032742(n)+A046523(n))^2) - A032742(n) - 3*A046523(n));
write_to_bfile(1,rgs_transform(vector(65537,n,A285729(n))),"b300229.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");

A300240 Filter sequence combining A009195(n) and A046523(n), i.e., gcd(n,phi(n)) and the prime signature of n.

Original entry on oeis.org

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

Views

Author

Antti Karttunen, Mar 02 2018

Keywords

Comments

Restricted growth sequence transform of P(A009195(n), A046523(n)), where P(a,b) is a two-argument form of A000027 used as a Cantor pairing function N x N -> N.

Examples

			a(6) = a(10) (= 4) because both 6 and 10 are nonsquare semiprimes, and A009195(6) = A009195(10) = 2.

Links

Crossrefs

Cf. A009195, A046523.
Cf. also A300230, A300241, A300242, A300243.

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])); }
A009195(n) = gcd(n, eulerphi(n));
A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); };  \\ From A046523
Aux300240(n) = (1/2)*(2 + ((A046523(n)+A009195(n))^2) - A046523(n) - 3*A009195(n));
write_to_bfile(1,rgs_transform(vector(65537,n,Aux300240(n))),"b300240.txt");

A355834 Lexicographically earliest infinite sequence such that a(i) = a(j) => A348717(i) = A348717(j) and A355931(i) = A355931(j) for all i, j >= 1.

Original entry on oeis.org

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

Views

Author

Antti Karttunen, Jul 20 2022

Keywords

Comments

Restricted growth sequence transform of the ordered pair [A348717(n), A355931(n)], where A355931(n) = A000265(A009194(i)).

Examples

			a(450) = a(3675) [= 274 as allotted by rgs-transform] because A003961(450) = 3675, therefore 450 and 3675 are in the same column of the prime shift array A246278, and because A355931(450) = A355931(3675) = 3.
a(3185) = a(14399) [= 2020 as allotted by rgs-transform] because A003961(3185) = 14399 and A355931(3185) = A355931(14399) = 7.
a(5005) = a(17017) [= 3184 as allotted by rgs-transform] because A003961(5005) = 17017 and A355931(5005) = A355931(17017) = 7.

Links

Crossrefs

Cf. A000203, A000265, A009194, A246278, A348717, A355931.
Cf. also A300230, A305800, A305897, A355833.

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));
A009194(n) = gcd(n, sigma(n));
A348717(n) = if(1==n, 1, my(f = factor(n), k = primepi(f[1, 1])-1); for (i=1, #f~, f[i, 1] = prime(primepi(f[i, 1])-k)); factorback(f));
Aux355834(n) = [A000265(A009194(n)), A348717(n)];
v355834 = rgs_transform(vector(up_to,n,Aux355834(n)));
A355834(n) = v355834[n];

A319337 Filter sequence combining gcd(n,tau(n)) (= A009191) with the prime signature of n (A046523).

Original entry on oeis.org

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

Views

Author

Antti Karttunen, Sep 24 2018

Keywords

Comments

Restricted growth sequence transform of ordered pair [A009191(n), A046523(n)].

Links

Crossrefs

Cf. A009191, A046523, A101296, A300230, A300240, A305801, A319338.

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; };
A009191(n) = gcd(n, numdiv(n));
A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p=0); prod(i=1, #f, (p=nextprime(p+1))^f[i]); };
v319337 = rgs_transform(vector(up_to,n,[A009191(n),A046523(n)]));
A319337(n) = v319337[n];

A325384 Lexicographically earliest sequence such that for all i, j, a(i) = a(j) => f(i) = f(j) where f(n) = [A000203(n), A009194(n)] for all other numbers, except f(p) = 0 for odd primes.

Original entry on oeis.org

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

Views

Author

Antti Karttunen, May 08 2019

Keywords

Links

Crossrefs

Cf. A000203, A009194, A300230, A305801, A325382, A325383.

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; };
A009194(n) = gcd(n,sigma(n));
Aux325384(n) = if((n%2)&&isprime(n),0,[sigma(n),A009194(n)]);
v325384 = rgs_transform(vector(up_to,n,Aux325384(n)));
A325384(n) = v325384[n];
Showing 1-9 of 9 results.

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