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 1-4 of 4 results.

A052126 a(1) = 1; for n>1, a(n)=n/(largest prime dividing n).

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 1, 4, 3, 2, 1, 4, 1, 2, 3, 8, 1, 6, 1, 4, 3, 2, 1, 8, 5, 2, 9, 4, 1, 6, 1, 16, 3, 2, 5, 12, 1, 2, 3, 8, 1, 6, 1, 4, 9, 2, 1, 16, 7, 10, 3, 4, 1, 18, 5, 8, 3, 2, 1, 12, 1, 2, 9, 32, 5, 6, 1, 4, 3, 10, 1, 24, 1, 2, 15, 4, 7, 6, 1, 16, 27, 2, 1, 12, 5, 2, 3, 8, 1, 18, 7, 4, 3, 2, 5, 32, 1
Offset: 1

Views

Author

James Sellers, Jan 21 2000

Keywords

Comments

For n>1, a(n)=1 if and only if n is prime. - Zak Seidov, Feb 09 2015
For n > 1, a(n) is the smallest divisor of n such that n/a(n) is prime. - David James Sycamore, Jan 03 2024

Examples

			a(15) = 15/(largest prime dividing 15) = 15/5 = 3.

Links

Crossrefs

Cf. A000040, A006530, A020639, A032742, A130065, A171462, A322820, A322826 (rgs-transform), A329697, A331410.
Left inverse of A253560.

Programs

  • Maple
    a := n -> `if`(n=1, 1, n/max(numtheory[factorset](n)));
seq(a(n), n=1..97); # Peter Luschny, Jul 28 2014
  • Mathematica
    a052126[n_] := Array[If[n == 1, 1, #/FactorInteger[#][[-1]][[1]]] &, n]; a052126[97] (* Michael De Vlieger, Dec 21 2014 *)
  • PARI
    gpf(n)=my(f=factor(n)[,1]); f[#f]
a(n)=if(n<4,return(1)); n/gpf(n) \\ Charles R Greathouse IV, Apr 28 2015

Formula

a(n) = n/A006530(n).
a(n) = A130065(n)/A020639(n). - Reinhard Zumkeller, May 05 2007
a(A002110(n)) = A002110(n-1), a(p^k) = p^(k-1), p any prime; k >= 1. - David James Sycamore, Jan 03 2024
a(n) = n - A171462(n). - Antti Karttunen, Jan 04 2024

A322865 a(n) = A000265(A122111(n)).

Original entry on oeis.org

1, 1, 1, 3, 1, 3, 1, 5, 9, 3, 1, 5, 1, 3, 9, 7, 1, 15, 1, 5, 9, 3, 1, 7, 27, 3, 25, 5, 1, 15, 1, 11, 9, 3, 27, 21, 1, 3, 9, 7, 1, 15, 1, 5, 25, 3, 1, 11, 81, 45, 9, 5, 1, 35, 27, 7, 9, 3, 1, 21, 1, 3, 25, 13, 27, 15, 1, 5, 9, 45, 1, 33, 1, 3, 75, 5, 81, 15, 1, 11, 49, 3, 1, 21, 27, 3, 9, 7, 1, 35, 81, 5, 9, 3, 27, 13, 1, 135, 25, 63, 1, 15, 1, 7, 75
Offset: 1

Views

Author

Antti Karttunen, Dec 30 2018

Keywords

Links

Crossrefs

Cf. A000265, A122111, A322813, A322819, A322820, A322826, A322865, A322867.

Programs

  • Mathematica
    Array[#/2^IntegerExponent[#, 2] &@ If[# < 3, 1, Block[{k = #, m = 0}, Times @@ Power @@@ Table[k -= m; k = DeleteCases[k, 0]; {Prime@ Length@ k, m = Min@ k}, Length@ Union@ k]] &@ Catenate[ConstantArray[PrimePi[#1], #2] & @@@ FactorInteger@ #]] &, 105] (* Michael De Vlieger, Dec 31 2018, after JungHwan Min at A122111 *)
  • PARI
    A000265(n) = (n>>valuation(n, 2));
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)};
A122111(n) = if(1==n,n,prime(bigomega(n))*A122111(A064989(n)));
A322865(n) = A000265(A122111(n));

Formula

a(n) = A000265(A122111(n)).
a(n) = A122111(A322820(n)).
A000005(a(n)) = A322813(n).
A000203(a(n)) = A322819(n).
A122111(a(n)) = A322820(n).
A000120(a(n)) = A322867(n).

A322820 a(n) = A052126(n) * A006530(A052126(n)).

Original entry on oeis.org

1, 1, 1, 4, 1, 4, 1, 8, 9, 4, 1, 8, 1, 4, 9, 16, 1, 18, 1, 8, 9, 4, 1, 16, 25, 4, 27, 8, 1, 18, 1, 32, 9, 4, 25, 36, 1, 4, 9, 16, 1, 18, 1, 8, 27, 4, 1, 32, 49, 50, 9, 8, 1, 54, 25, 16, 9, 4, 1, 36, 1, 4, 27, 64, 25, 18, 1, 8, 9, 50, 1, 72, 1, 4, 75, 8, 49, 18, 1, 32, 81, 4, 1, 36, 25, 4, 9, 16, 1, 54, 49, 8, 9, 4, 25, 64, 1, 98, 27, 100, 1, 18, 1, 16, 75
Offset: 1

Views

Author

Antti Karttunen, Dec 27 2018

Keywords

Links

Crossrefs

Cf. A000265, A006530, A052126, A070003 (positions where a(n) = n for n > 1), A122111, A319988, A322813, A322819, A322826 (restricted growth sequence transform).

Programs

Formula

a(n) = A052126(n) * A006530(A052126(n)).
a(n) = A122111(A000265(A122111(n))).
A052126(a(n)) = A052126(n).
a(n) <= n.
For all n > 1, A010051(n) + A319988(a(n)) = 1.

A339874 Lexicographically earliest infinite sequence such that a(i) = a(j) => f(i) = f(j), where f(n) = A052126(n) for n > 1, and f(1) = 0.

Original entry on oeis.org

1, 2, 2, 3, 2, 3, 2, 4, 5, 3, 2, 4, 2, 3, 5, 6, 2, 7, 2, 4, 5, 3, 2, 6, 8, 3, 9, 4, 2, 7, 2, 10, 5, 3, 8, 11, 2, 3, 5, 6, 2, 7, 2, 4, 9, 3, 2, 10, 12, 13, 5, 4, 2, 14, 8, 6, 5, 3, 2, 11, 2, 3, 9, 15, 8, 7, 2, 4, 5, 13, 2, 16, 2, 3, 17, 4, 12, 7, 2, 10, 18, 3, 2, 11, 8, 3, 5, 6, 2, 14, 12, 4, 5, 3, 8, 15, 2, 19, 9, 20, 2, 7, 2, 6, 17
Offset: 1

Views

Author

Antti Karttunen, Dec 25 2020

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) => A322826(i) = A322826(j).

Links

Crossrefs

Cf. A052126, A322826.
Cf. also A305800, A300226, A334107, A334108.

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; };
A052126(n) = if(1==n,n,(n/vecmax(factor(n)[, 1])));
Aux339874(n) = if(1==n,0,A052126(n));
v339874 = rgs_transform(vector(up_to, n, Aux339874(n)));
A339874(n) = v339874[n];

Formula

a(1) = 1; for n > 1, a(n) = 1 + A322826(n).
Showing 1-4 of 4 results.

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