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.

A322587 Lexicographically earliest such sequence a that for all i, j, a(i) = a(j) => f(i) = f(j), where f(n) = 0 for odd primes, and f(n) = A291756(n) [equally: A295887(n)] for any other number.

Original entry on oeis.org

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

Views

Author

Antti Karttunen, Dec 18 2018

Keywords

Comments

For all i, j: a(i) = a(j) => A322320(i) = A322320(j).

Links

Crossrefs

Cf. A291756, A295887, A322588, A322589, A322591, A322592.

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; };
A003557(n) = { my(f=factor(n)); for(i=1, #f~, f[i, 2] = f[i, 2]-1); factorback(f); };
A173557(n) = my(f=factor(n)[, 1]); prod(k=1, #f, f[k]-1); \\ From A173557
Aux322587(n) = if((n>2)&&isprime(n),0,(1/2)*(2 + ((A003557(n)+A173557(n))^2) - A003557(n) - 3*A173557(n)));
v322587 = rgs_transform(vector(up_to, n, Aux322587(n)));
A322587(n) = v322587[n];

A322321 a(n) = lcm(A003557(n), A173557(n)).

Original entry on oeis.org

1, 1, 2, 2, 4, 2, 6, 4, 6, 4, 10, 2, 12, 6, 8, 8, 16, 6, 18, 4, 12, 10, 22, 4, 20, 12, 18, 6, 28, 8, 30, 16, 20, 16, 24, 6, 36, 18, 24, 4, 40, 12, 42, 10, 24, 22, 46, 8, 42, 20, 32, 12, 52, 18, 40, 12, 36, 28, 58, 8, 60, 30, 12, 32, 48, 20, 66, 16, 44, 24, 70, 12, 72, 36, 40, 18, 60, 24, 78, 8, 54, 40, 82, 12, 64, 42, 56, 20
Offset: 1

Views

Author

Antti Karttunen, Dec 05 2018

Keywords

Links

Crossrefs

Cf. A000010, A003557, A173557, A322320, A322351, A322352.
Cf. also A322319, A322359.

Programs

  • Mathematica
    a[n_] := If[n == 1, 1, Module[{f=FactorInteger[n]}, LCM[ Times@@ (First[#] ^(Last[#]-1)& /@  f), Times@@((#-1)& @@@ f)]]]; Array[a, 120] (* Amiram Eldar, Dec 05 2018 *)
  • PARI
    A003557(n) = { my(f=factor(n)); for (i=1, #f~, f[i, 2] = f[i, 2]-1); factorback(f); }; \\ From A003557
A173557(n) = factorback(apply(p -> p-1, factor(n)[, 1]));
A322321(n) = lcm(A003557(n), A173557(n));

Formula

a(n) = lcm(A003557(n), A173557(n)) = lcm(A322351(n), A322352(n)).
a(n) = A000010(n) / A322320(n).

A322318 a(n) = gcd(A003557(n), A048250(n)).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 3, 1, 2, 1, 1, 1, 4, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 6, 1, 1, 1, 2, 1, 1, 1, 2, 3, 1, 1, 4, 1, 1, 1, 2, 1, 3, 1, 4, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 12, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 4, 1, 3, 1, 2, 1, 1, 1, 4, 1, 1, 3, 2, 1, 1, 1, 2, 1
Offset: 1

Views

Author

Antti Karttunen, Dec 05 2018

Keywords

Links

Crossrefs

Cf. A001615, A003557, A048250, A322319.
Cf. also A066086, A322320.

Programs

  • Mathematica
    a[n_] := If[n == 1, 1, Module[{f=FactorInteger[n]}, GCD[ Times@@ (First[#] ^(Last[#]-1)& /@  f), Times@@((#+1)& @@@ f)]]]; Array[a, 120] (* Amiram Eldar, Dec 05 2018 *)
  • PARI
    A003557(n) = { my(f=factor(n)); for(i=1, #f~, f[i, 2] = f[i, 2]-1); factorback(f); }; \\ From A003557
A048250(n) = factorback(apply(p -> p+1, factor(n)[, 1]));
A322318(n) = gcd(A048250(n), A003557(n));

Formula

a(n) = gcd(A003557(n), A048250(n)).
a(n) = A001615(n) / A322319(n).

A322351 a(n) = min(A003557(n), A173557(n)).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 2, 4, 1, 2, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 4, 1, 1, 1, 2, 3, 1, 1, 2, 6, 4, 1, 2, 1, 2, 1, 4, 1, 1, 1, 2, 1, 1, 3, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 5, 2, 1, 1, 1, 4, 2, 1, 1, 2, 1, 1, 1, 4, 1, 3, 1, 2, 1, 1, 1, 2, 1, 6, 3, 4, 1, 1, 1, 4, 1
Offset: 1

Views

Author

Antti Karttunen, Dec 05 2018

Keywords

Links

Crossrefs

Cf. A000010, A003557, A173557, A322320, A322321, A322352, A322355.

Programs

  • Mathematica
    a[n_] := If[n == 1, 1, Module[{f=FactorInteger[n]}, Min[ Times@@ (First[#]^(Last[#]-1)& /@  f), Times@@((#-1)& @@@ f)]]]; Array[a, 120] (* Amiram Eldar, Dec 05 2018 *)
  • PARI
    A003557(n) = { my(f=factor(n)); for (i=1, #f~, f[i, 2] = f[i, 2]-1); factorback(f); }; \\ From A003557
A173557(n) = factorback(apply(p -> p-1, factor(n)[, 1]));
A322351(n) = min(A003557(n), A173557(n));

Formula

a(n) = min(A003557(n), A173557(n)).
a(n) = A000010(n) / A322352(n).

A322352 a(n) = max(A003557(n), A173557(n)).

Original entry on oeis.org

1, 1, 2, 2, 4, 2, 6, 4, 3, 4, 10, 2, 12, 6, 8, 8, 16, 3, 18, 4, 12, 10, 22, 4, 5, 12, 9, 6, 28, 8, 30, 16, 20, 16, 24, 6, 36, 18, 24, 4, 40, 12, 42, 10, 8, 22, 46, 8, 7, 5, 32, 12, 52, 9, 40, 6, 36, 28, 58, 8, 60, 30, 12, 32, 48, 20, 66, 16, 44, 24, 70, 12, 72, 36, 8, 18, 60, 24, 78, 8, 27, 40, 82, 12, 64, 42, 56, 10, 88, 8
Offset: 1

Views

Author

Antti Karttunen, Dec 05 2018

Keywords

Links

Crossrefs

Cf. A000010, A003557, A173557, A322320, A322321, A322351, A322355.

Programs

  • Mathematica
    a[n_] := If[n == 1, 1, Module[{f=FactorInteger[n]}, Max[ Times @@ (First[#]^ (Last[#]-1)& /@  f), Times@@((#-1)& @@@ f)]]]; Array[a, 120] (* Amiram Eldar, Dec 05 2018 *)
  • PARI
    A003557(n) = { my(f=factor(n)); for (i=1, #f~, f[i, 2] = f[i, 2]-1); factorback(f); }; \\ From A003557
A173557(n) = factorback(apply(p -> p-1, factor(n)[, 1]));
A322352(n) = max(A003557(n), A173557(n));

Formula

a(n) = max(A003557(n), A173557(n)).
a(n) = A000010(n) / A322351(n).

A322355 Lexicographically earliest such sequence a that a(i) = a(j) => A322351(i) = A322351(j) and A322352(i) = A322352(j), for all i, j.

Original entry on oeis.org

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

Views

Author

Antti Karttunen, Dec 05 2018

Keywords

Comments

Restricted growth sequence transform of the ordered pair [A322351(n), A322352(n)].
Essentially also the restricted growth sequence transform of the unordered pair {A003557(n), A173557(n)}.
For all i, j:
A295887(i) = A295887(j) => a(i) = a(j),
a(i) = a(j) => A322320(i) = A322320(j),
a(i) = a(j) => A322321(i) = A322321(j),
a(i) = a(j) => A000010(i) = A000010(j).

Links

Crossrefs

Cf. A000010, A003557, A173557, A295887, A322320, A322321, A322351, A322352.

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; };
A003557(n) = { my(f=factor(n)); for (i=1, #f~, f[i, 2] = f[i, 2]-1); factorback(f); }; \\ From A003557
A173557(n) = factorback(apply(p -> p-1, factor(n)[, 1]));
A322351(n) = min(A003557(n), A173557(n));
A322352(n) = max(A003557(n), A173557(n));
v322355 = rgs_transform(vector(up_to, n, [A322351(n), A322352(n)]));
A322355(n) = v322355[n];
Showing 1-6 of 6 results.

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