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.

A137926 a(n) = the largest divisor of n that is coprime to A000005(n). (A000005(n) = the number of positive divisors of n.)

Original entry on oeis.org

1, 1, 3, 4, 5, 3, 7, 1, 1, 5, 11, 1, 13, 7, 15, 16, 17, 1, 19, 5, 21, 11, 23, 3, 25, 13, 27, 7, 29, 15, 31, 1, 33, 17, 35, 4, 37, 19, 39, 5, 41, 21, 43, 11, 5, 23, 47, 3, 49, 25, 51, 13, 53, 27, 55, 7, 57, 29, 59, 5, 61, 31, 7, 64, 65, 33, 67, 17, 69, 35, 71, 1, 73, 37, 25, 19, 77, 39, 79
Offset: 1

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Author

Leroy Quet, Feb 23 2008

Keywords

Examples

			6 has 4 positive divisors. The divisors of 6 are 1,2,3,6. The divisors of 6 that are coprime to 4 are 1 and 3. 3 is the largest of these; so a(6) = 3.

Links

Crossrefs

Cf. A046642 (a(n)=n), A120737 (a(n)=1), A137927.

Programs

  • Maple
    f := proc (n) local D, t; D := numtheory:-divisors(n); t := nops(D); max(select(proc (d) options operator, arrow; igcd(d, t) = 1 end proc, D)) end proc:
map(f, [$1..100]); # Robert Israel, Feb 11 2018
  • Mathematica
    Table[Select[Divisors[n], GCD[ #, Length[Divisors[n]]] == 1 &][[ -1]], {n, 1, 80}] (* Stefan Steinerberger, Mar 09 2008 *)
  • PARI
    a(n) = {my(d = divisors(n)); vecmax(select(x->(gcd(x, #d) == 1), d));} \\ Michel Marcus, Feb 12 2018

Extensions

More terms from Stefan Steinerberger, Mar 09 2008

A291186 a(n) = numerator of (pod(n) / tau(n)).

Original entry on oeis.org

1, 1, 3, 8, 5, 9, 7, 16, 9, 25, 11, 288, 13, 49, 225, 1024, 17, 972, 19, 4000, 441, 121, 23, 41472, 125, 169, 729, 10976, 29, 101250, 31, 16384, 1089, 289, 1225, 1119744, 37, 361, 1521, 320000, 41, 388962, 43, 42592, 30375, 529, 47, 127401984, 343, 62500, 2601
Offset: 1

Views

Author

Jaroslav Krizek, Sep 05 2017

Keywords

Comments

pod(n) = the product of the divisors of n (A007955), tau(n) = the number of the divisors of n (A000005).

Examples

			For n = 4; pod(4) / tau(4) = 8 / 3; a(n) = 8.

Links

Crossrefs

Cf. A137927 (denominator).
Cf. A000005, A007955, A120736.

Programs

  • Magma
    [Numerator(&*[d: d in Divisors(n)] / #[d: d in Divisors(n)]): n in [1..1000]];
  • Maple
    f:= proc(n) local D; D:= numtheory:-divisors(n); numer(convert(D,`*`)/nops(D)) end proc:
map(f, [$1..100]); # Robert Israel, Sep 14 2017
  • Mathematica
    Table[Numerator[Apply[Times, Divisors@ n]/DivisorSigma[0, n]], {n, 51}] (* Michael De Vlieger, Sep 05 2017 *)
  • PARI
    a(n) = my(d=divisors(n)); numerator(prod(k=1, #d, d[k])/#d); \\ Michel Marcus, Sep 05 2017

Formula

a(n) = numerator of (A007955(n) / A000005(n)).

A291899 Numbers n such that (pod(n)/tau(n)) > (pod(k)/tau(k)) for all k < n.

Original entry on oeis.org

1, 3, 4, 6, 8, 10, 12, 18, 20, 24, 30, 36, 48, 60, 72, 84, 90, 96, 108, 120, 168, 180, 240, 336, 360, 420, 480, 504, 540, 600, 630, 660, 672, 720, 840, 1080, 1260, 1440, 1680, 2160, 2520, 3360, 3780, 3960, 4200, 4320, 4620, 4680, 5040, 7560, 9240, 10080, 12600
Offset: 1

Views

Author

Jaroslav Krizek, Oct 10 2017

Keywords

Comments

pod(n) = the product of the divisors of n (A007955), tau(n) = the number of the divisors of n (A000005).
Contains all members of A002182 except 2. - Robert Israel, Nov 09 2017
Is this the same as A034288 except for 3? - Georg Fischer, Oct 09 2018
From David A. Corneth, Oct 11 2018: (Start)
Various methods exist to find terms for this sequence, possibly combinable:
- Brute force; checking every positive integer up to some bound.
- Finding terms based on the prime signature.
- Relating to that, the number of divisors.
- Finding terms based on the GCD of some earlier found terms.
- ... (?)
There seems to be a method that helps finding terms < 10^150 for the similar A034287. (End)

Examples

			6 is a term because pod(6)/tau(6) = 36/4 = 9 > pod(k)/tau(k) for all k < 6.

Links

Crossrefs

Cf. A000005, A002182, A034287, A034288, A007955, A120736, A137927, A291186, A067128.

Programs

  • Magma
    a:=1; S:=[a]; for n in [2..60] do k:=0; flag:= true; while flag do k+:=1; if &*[d: d in Divisors(a)] / #[d: d in Divisors(a)] lt &*[d: d in Divisors(k)] / #[d: d in Divisors(k)] then Append(~S, k); a:=k; flag:=false; end if; end while; end for; S;
  • Maple
    f:= proc(n) local t; t:= numtheory:-tau(n); simplify(n^(t/2))/t end proc:
N:= 20000: # to get all terms <= N
Res:= NULL: m:= 0:
for n from 1 to N do
  v:= f(n);
  if v > m then Res:= Res, n; m:= v fi
od:
Res; # Robert Israel, Nov 09 2017
  • Mathematica
    With[{s = Array[Times @@ Divisors@ # &, 12600]}, Select[Range@ Length@ s, Function[m, AllTrue[Range[# - 1], m > s[[#]]/DivisorSigma[0, #] &]][s[[#]]/DivisorSigma[0, #]] &]] (* Michael De Vlieger, Oct 10 2017 *)
DeleteDuplicates[Table[{n,Times@@Divisors[n]/DivisorSigma[0,n]},{n,13000}],GreaterEqual[ #1[[2]],#2[[2]]]&][[;;,1]] (* Harvey P. Dale, Mar 03 2024 *)

Formula

Numbers n such that (A007955(n)/A000005(n)) > (A007955(k)/A000005(k)) for all k < n.
Numbers n such that (A291186(n)/A137927(n)) > (A291186(k)/A137927(k)) for all k < n.

A293376 Corresponding values of pod(n)/tau(n) of numbers n from A120736.

Original entry on oeis.org

1, 1, 9, 16, 9, 25, 288, 49, 972, 121, 41472, 169, 101250, 289, 1119744, 361, 320000, 388962, 529, 1062882, 1229312, 841, 3888000000, 961, 2371842, 3001250, 11609505792, 1369, 4626882, 327680000, 1681, 29274835968, 1849, 7496192, 44286750000, 2209, 65229815808
Offset: 1

Views

Author

Jaroslav Krizek, Oct 07 2017

Keywords

Comments

Integer of pod(n)/tau(n) of numbers n such that tau(n) = the number of the divisors of n (A000005) divides pod(n) = the product of the divisors of n (A007955).

Examples

			For n = 3; A120736(3) = 6; pod(6)/tau(6) = 36/4 = 9.

Links

Crossrefs

Cf. A000005, A007955, A120736, A137927, A291186.

Programs

  • Magma
    [Numerator(&*[d: d in Divisors(n)] / #[d: d in Divisors(n)]): n in [1..1000] | Denominator(&*[d: d in Divisors(n)] / #[d: d in Divisors(n)]) eq 1];
  • Mathematica
    (Times @@ #)/Length@ # &@ Divisors@ # & /@ Select[Range@ 100, Divisible[Times @@ Divisors@ #, DivisorSigma[0, #]] &] (* Michael De Vlieger, Oct 10 2017 *)

Formula

a(n) = A291186(A120736(n))/A137927(A120736(n)).
Showing 1-4 of 4 results.

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