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 1-3 of 3 results.

A300249 Filter sequence combining A003415(n) and A046523(n), the arithmetic derivative of n 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
Offset: 1

Views

Author

Antti Karttunen, Mar 04 2018

Keywords

Comments

Restricted growth sequence transform of P(A003415(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(51) = a(91) (= 33) because both are nonsquare semiprimes (3*17 and 7*13), and also their arithmetic derivatives are equal, as A003415(51) = A003415(91) = 20.
a(78) = a(105) (= 56) because both have the same prime signature (78 = 2*3*13 and 105 = 3*5*7), and also their arithmetic derivatives are equal, as A003415(78) = A003415(105) = 71.

Links

Crossrefs

Cf. A003415, A046523.
Cf. also A300226, A300229, A300248.
Differs from A300235 for the first time at n=105, where a(105)=56, while A300235(105)=75.

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])); }
A003415(n) = {my(fac); if(n<1, 0, fac=factor(n); sum(i=1, matsize(fac)[1], n*fac[i, 2]/fac[i, 1]))}; \\ From A003415
A046523(n) = my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]) \\ From A046523
Aux300249(n) = ((1/2)*(2 + ((A003415(n)+A046523(n))^2) - A003415(n) - 3*A046523(n)));
write_to_bfile(1,rgs_transform(vector(65537,n,Aux300248(n))),"b300249.txt");

A300247 Restricted growth sequence transform of A286457(n), filter combining A078898(n) and A246277(n).

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

Views

Author

Antti Karttunen, Mar 03 2018

Keywords

Comments

For all i, j:
a(i) = a(j) => A280492(i) = A280492(j).
a(i) = a(j) => A300248(i) = A300248(j).
The latter follows because A046523(n) = A046523(2*A246277(n)).

Examples

			a(65) = a(119) (= 42) because A078898(65) = A078898(119) = 5 (both numbers occur in column 5 of A083221) and because A246277(65) = A246277(119) = 7 (both numbers occur in column 7 of A246278). Note that 65 = 5*13 = prime(3)*prime(6) and 119 = 7*17 = prime(4)*prime(7) = A003961(65). A246277(n) contains complete information about the (relative) differences between prime indices in the prime factorization of n.

Links

Crossrefs

Cf. A078898, A246277, A286457.
Cf. also A300248, A280492.

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])); }
A020639(n) = { if(1==n,n,vecmin(factor(n)[, 1])); };
A064989(n) = {my(f); f = factor(n); if((n>1 && f[1,1]==2), f[1,2] = 0); for (i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f)};
A078898(n) = { if(n<=1,n, my(spf=A020639(n),k=1,m=n/spf); while(m>1,if(A020639(m)>=spf,k++); m--); (k)); }; \\ Antti Karttunen, Mar 03 2018
A246277(n) = { if(1==n, 0, while((n%2), n = A064989(n)); (n/2)); };
A286457(n) = if(1==n,0,(1/2)*(2 + ((A078898(n)+A246277(n))^2) - A078898(n) - 3*A246277(n)));
write_to_bfile(1,rgs_transform(vector(65537,n,A286457(n))),"b300247.txt");

A300246 Filter sequence combining A046523(n) and A078899(n), the prime signature of n and the number of times the greatest prime factor of n is the greatest prime factor for numbers <= 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, 8, 4, 2, 11, 12, 4, 13, 7, 2, 14, 2, 15, 8, 4, 16, 17, 2, 4, 8, 18, 2, 14, 2, 7, 19, 4, 2, 20, 21, 22, 8, 7, 2, 23, 16, 24, 8, 4, 2, 25, 2, 4, 22, 26, 16, 14, 2, 7, 8, 27, 2, 28, 2, 4, 29, 7, 30, 14, 2, 31, 32, 4, 2, 33, 16, 4, 8, 24, 2, 34, 30, 7, 8, 4, 16, 35, 2, 36, 22, 37, 2, 14, 2, 24, 38
Offset: 1

Views

Author

Antti Karttunen, Mar 09 2018

Keywords

Comments

Restricted growth sequence transform of P(A046523(n), A078899(n)), where P(a,b) is a two-argument form of A000027 used as a Cantor pairing function N x N -> N.

Examples

			a(30) = a(42) (= 14) because A078899(30) = A078899(42) = 6 and both numbers are products of three distinct primes, thus have the same prime signature.
a(35) = a(55) = a(65) (= 16) because A078899(35) = A078899(55) = A078899(65) = 5 and because all three are nonsquare semiprimes.

Links

Crossrefs

Cf. A046523, A078899.
Cf. also A300247, A300248.
Differs from A300226 for the first time at n=40, where a(40) = 18.

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])); }
A006530(n) = if(1==n, n, vecmax(factor(n)[, 1]));
A078899(n) = { if(n<=1,n, my(gpf=A006530(n),k=1,m=n/gpf); while(m>1,if(A006530(m)<=gpf,k++); m--); (k)); };
A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); };  \\ From A046523
Aux300246(n) = if(1==n,0,(1/2)*(2 + ((A078899(n)+A046523(n))^2) - A078899(n) - 3*A046523(n)));
write_to_bfile(1,rgs_transform(vector(up_to,n,Aux300246(n))),"b300246.txt");
Showing 1-3 of 3 results.

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