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.

A305800 Filter sequence for a(prime) = constant sequences.

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, 40, 41, 42, 43, 2, 44, 2, 45, 46, 47, 48, 49, 2, 50, 51, 52, 2, 53, 2, 54, 55, 56, 57, 58, 2, 59, 60, 61, 2, 62, 63, 64, 65, 66, 2, 67, 68, 69, 70, 71, 72, 73, 2, 74, 75, 76, 2, 77, 2, 78, 79, 80, 2, 81, 2, 82, 83, 84, 2, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96
Offset: 1

Views

Author

Antti Karttunen, Jun 14 2018

Keywords

Comments

Restricted growth sequence transform of A239968.
In the following, A stands for this sequence, A305800, and S -> T (where S and T are sequence A-numbers) indicates that for all i, j: S(i) = S(i) => T(i) = T(j).
For example, the following implications hold:
A -> A300247 -> A305897 -> A077462 -> A101296,
A -> A290110 -> A300250 -> A101296.

Links

Crossrefs

Cf. A000720, A239968.
Differs from A296073 for the first time at n=125, as a(125) = 96, while A296073(125) = 33.
Cf. also A305900, A305801, A295300, A289626 for other "upper level" filters.

Programs

  • Mathematica
    Join[{1},Table[If[PrimeQ[n],2,1+n-PrimePi[n]],{n,2,150}]] (* Harvey P. Dale, Jul 12 2019 *)
  • PARI
    A305800(n) = if(1==n,n,if(isprime(n),2,1+n-primepi(n)));

Formula

a(1) = 1; for n > 1, a(n) = 2 for prime n, and a(n) = 1+n-A000720(n) for composite n.

A305897 Lexicographically earliest infinite sequence such that a(i) = a(j) => A348717(i) = A348717(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, 8, 18, 4, 19, 2, 20, 12, 21, 2, 22, 2, 23, 7, 24, 2, 25, 3, 26, 14, 27, 2, 28, 6, 29, 18, 30, 2, 31, 2, 32, 11, 33, 8, 34, 2, 35, 20, 36, 2, 37, 2, 38, 10, 39, 4, 40, 2, 41, 9, 42, 2, 43, 12, 44, 24, 45, 2, 46, 6, 47, 30, 48, 14, 49, 2, 50, 15, 51, 2, 52, 2, 53, 16
Offset: 1

Views

Author

Antti Karttunen, Jun 14 2018

Keywords

Comments

Restricted growth sequence transform of A348717, or equally, of A246277.
Filter sequence for prime factorization patterns, including also information about gaps between prime factors. - Original name, gives the motivation for this sequence. Here the "gaps" refers to differences between the indices of primes present, not the prime gaps as usually understood.
For all i, j:
A305800(i) = A305800(j) => a(i) = a(j),
a(i) = a(j) => A077462(i) = A077462(j) => A101296(i) = A101296(j).
a(i) = a(j) => A243055(i) = A243055(j).

Examples

			a(10) = a(21) (= 6) because both have prime exponents [1, 1] and the difference between the prime indices is the same, as 10 = prime(1)*prime(3), and 21 = prime(2)*prime(4).
a(12) != a(18) because the prime exponents [2,1] and [1,2] do not occur in the same order.
a(140) = a(693) (= 71) because both numbers have prime exponents [2, 1, 1] (in this order) and the differences between the indices of the successive prime factors are same: 140 = prime(1)^2 * prime(3) * prime(4), 693 = prime(2)^2 * prime(4) * prime(5).

Links

Crossrefs

Cf. A077462, A101296, A246277, A300247, A305800, A348717.
Cf. also A286621.

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; };
A246277(n) = if(1==n, 0, 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)/2);
v305897 = rgs_transform(vector(up_to,n,A246277(n)));
A305897(n) = v305897[n];

Extensions

Name changed by Antti Karttunen, Apr 30 2022

A323888 Lexicographically earliest sequence such that for all i, j, a(i) = a(j) => f(i) = f(j) where f(n) = [A032742(n),A302042(n)] for all n > 1, with f(1) = 0.

Original entry on oeis.org

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

Views

Author

Antti Karttunen, Feb 08 2019

Keywords

Comments

For all i, j:
A305800(i) = A305800(j) => a(i) = a(j),
a(i) = a(j) => A001222(i) = A001222(j),
a(i) = a(j) => A253557(i) = A253557(j).

Links

Crossrefs

Cf. A032742, A302042.
Cf. also A001222, A253557, A300247, A305800.

Programs

  • PARI
    up_to = 65537;
ordinal_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), pt); for(i=1, length(invec), if(mapisdefined(om,invec[i]), pt = mapget(om, invec[i]), pt = 0); outvec[i] = (1+pt); mapput(om,invec[i],(1+pt))); outvec; };
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; };
A020639(n) = if(n>1, if(n>n=factor(n, 0)[1, 1], n, factor(n)[1, 1]), 1); \\ From A020639
A032742(n) = (n/A020639(n));
v078898 = ordinal_transform(vector(up_to,n,A020639(n)));
A078898(n) = v078898[n];
A302042(n) = if((1==n)||isprime(n),1,my(c = A078898(n), p = prime(-1+primepi(A020639(n))+primepi(A020639(c))), d = A078898(c), k=0); while(d, k++; if((1==k)||(A020639(k)>=p),d -= 1)); (k*p));
A323888aux(n) = if(1==n, 0, [A032742(n),A302042(n)]);
v323888 = rgs_transform(vector(up_to, n, A323888aux(n)));
A323888(n) = v323888[n];

A300248 Filter sequence combining A046523(n) and A078898(n), the prime signature of n and the number of times the smallest prime factor of n is the smallest prime factor for numbers <= 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, 8, 24, 2, 25, 2, 26, 27, 28, 2, 29, 3, 30, 31, 32, 2, 33, 12, 34, 35, 36, 2, 37, 2, 38, 39, 40, 6, 41, 2, 42, 43, 44, 2, 45, 2, 46, 47, 48, 4, 49, 2, 50, 51, 52, 2, 53, 20, 54, 55, 56, 2, 57, 12, 58, 59, 60, 8, 61, 2, 62, 63, 64, 2
Offset: 1

Views

Author

Antti Karttunen, Mar 03 2018

Keywords

Comments

Restricted growth sequence transform of P(A046523(n), A078898(n)), where P(a,b) is a two-argument form of A000027 used as a Cantor pairing function N x N -> N.

Examples

			a(10) = a(65) (= 6) because A078898(10) = A078898(65) = 5 (both numbers occur in column 5 of A083221) and because both have the same prime-signature (both are nonsquare semiprimes).

Links

Crossrefs

Cf. A046523, A078898.
Cf. also A300226, A300229, A300247.

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])); };
A046523(n) = my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]) \\ From A046523
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)); };
Aux300248(n) = if(1==n,0,(1/2)*(2 + ((A078898(n)+A046523(n))^2) - A078898(n) - 3*A046523(n)));
write_to_bfile(1,rgs_transform(vector(65537,n,Aux300248(n))),"b300248.txt");

A319339 Filter sequence combining A081373(n) with A246277(n).

Original entry on oeis.org

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

Views

Author

Antti Karttunen, Sep 24 2018

Keywords

Comments

Restricted growth sequence transform of ordered pair [A081373(n), A246277(n)].

Links

Crossrefs

Cf. A081373, A246277, A300247, A305801, A305897, A318500.

Programs

  • PARI
    up_to = 65537;
ordinal_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), pt); for(i=1, length(invec), if(mapisdefined(om,invec[i]), pt = mapget(om, invec[i]), pt = 0); outvec[i] = (1+pt); mapput(om,invec[i],(1+pt))); outvec; };
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; };
v081373 = ordinal_transform(vector(up_to,n,eulerphi(n)));
A081373(n) = v081373[n];
A246277(n) = if(1==n, 0, 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)/2);
v319339 = rgs_transform(vector(up_to,n,[A081373(n),A246277(n)]));
A319339(n) = v319339[n];

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 1-6 of 6 results.

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