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.

A056867 Nilpotent numbers: n such that every group of order n is nilpotent.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 9, 11, 13, 15, 16, 17, 19, 23, 25, 27, 29, 31, 32, 33, 35, 37, 41, 43, 45, 47, 49, 51, 53, 59, 61, 64, 65, 67, 69, 71, 73, 77, 79, 81, 83, 85, 87, 89, 91, 95, 97, 99, 101, 103, 107, 109, 113, 115, 119, 121, 123, 125, 127, 128, 131, 133, 135, 137, 139
Offset: 1

Views

Author

N. J. A. Sloane, Sep 02 2000

Keywords

Comments

Contains exactly the numbers n for which gcd(n,|A153038(n)|)=1 [Pazderski]. - R. J. Mathar, Apr 03 2012
A group G of order m is nilpotent iff it has a quotient group of order m/d for each divisor d of m. - Des MacHale and Bernard Schott, Jul 15 2022

Links

Crossrefs

Complement of A056868.
Cf. A003277, A051532, A056866.
Cf. A027748, A124010, A173557.

Programs

  • GAP
    IsNilpotentInt := function(n)
  local f, i, j; f := PrimePowersInt(n);
  for i in [1..Length(f)/2] do
    for j in [1..f[2*i]] do
      if Gcd(f[2*i-1]^j-1, n) > 1 then return false; fi;
    od;
  od;
  return true;
end;
Filtered([1..140], IsNilpotentInt); # Gheorghe Coserea, Dec 02 2017
  • Mathematica
    A153038[1] = 1; A153038[n_] := (x = 1; Do[p = f[[1]]; e = f[[2]]; x = x*Product[1 - p^s, {s, 1, e}], {f, FactorInteger[n]}]; x); A056867 = Select[Range[140], GCD[#, Abs[A153038[#]]] == 1 &] (* Jean-François Alcover, May 15 2012, after R. J. Mathar *)
  • PARI
    is(n)=my(f=factor(n));for(k=1,#f[,1], for(j=1,f[k,2], if(gcd(n, f[k,1]^j-1)>1, return(0)))); 1 \\ Charles R Greathouse IV, Sep 18 2012

Formula

n is in this sequence if p^k is not congruent to 1 mod q for any primes p and q dividing n such that p^e but not p^(e+1) divides n and k <= e. - Charles R Greathouse IV, Aug 27 2012

Extensions

More terms from Francisco Salinas (franciscodesalinas(AT)hotmail.com), Dec 25 2001

A183091 a(n) is the product of the divisors p^k of n where p is prime and k >= 1.

Original entry on oeis.org

1, 2, 3, 8, 5, 6, 7, 64, 27, 10, 11, 24, 13, 14, 15, 1024, 17, 54, 19, 40, 21, 22, 23, 192, 125, 26, 729, 56, 29, 30, 31, 32768, 33, 34, 35, 216, 37, 38, 39, 320, 41, 42, 43, 88, 135, 46, 47, 3072, 343, 250, 51, 104, 53, 1458
Offset: 1

Views

Author

Jaroslav Krizek, Dec 25 2010

Keywords

Comments

Product of n-th row of triangle A210208. - Reinhard Zumkeller, Mar 18 2012

Examples

			For n = 12, set of such divisors is {1, 2, 3, 4}; a(12) = 1*2*3*4 = 24.

Links

Crossrefs

Cf. A007955, A023888, A153038, A183092, A210208.

Programs

  • Haskell
    a183091 = product . a210208_row  -- Reinhard Zumkeller, Mar 18 2012
  • Maple
    A183091 := proc(n) local a,d; a := 1 ;for d in numtheory[divisors](n) minus {1} do  if nops( numtheory[factorset](d)) = 1 then a := a*d; end if; end do: a ; end proc: # R. J. Mathar, Apr 14 2011
  • Mathematica
    Table[Product[d, {d, Select[Divisors[n], Length[FactorInteger[#]] == 1 &]}], {n,1, 54}] (* Geoffrey Critzer, Mar 18 2015 *)
f[p_, e_] := p^(e*(e+1)/2); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Aug 31 2023 *)
  • PARI
    a(n)=my(f=factor(n)); prod(i=1,#f~, f[i,1]^binomial(f[i,2]+1,2)) \\ Charles R Greathouse IV, Nov 11 2014

Formula

a(n) = A007955(n) / A183092(n).
Multiplicative with a(p^k) = p^(k*(k+1)/2).
The Dirichlet g.f. of a(n) / abs(A153038(n)) is Product_{k >= 0} zeta(s+k). - Álvar Ibeas, Nov 10 2014

A181941 Numbers n such that some group of order n has a non-cyclic commutator group.

Original entry on oeis.org

12, 18, 24, 32, 36, 48, 50, 54, 56, 60, 64, 72, 75, 80, 81, 84, 90, 96, 98, 100, 108, 112, 120, 126, 128, 132, 144, 147, 150, 156, 160, 162, 168, 180, 192, 196, 198, 200, 204, 216, 224, 225, 228, 234, 240, 242, 243, 250, 252, 256, 264, 270, 276, 280, 288, 294, 300, 306, 312, 320, 324, 336, 338, 342
Offset: 1

Views

Author

R. J. Mathar, Apr 03 2012

Keywords

Comments

The complementary sequence 1, 2, 3, 4... is much denser and contains all n such that each group of order n has a cyclic commutator group.
Let the factorization of n into powers of squarefree mutually coprime numbers n_1, n_2, n_3, n_4, n_5,... be n = n_1 *n_2^2 *n_3^3 *n_4^4 * n_5^5*..., see A051903.
Then the complementary sequence contains n of the form n = n_1*n_2^2*n_3^3*n_4^4 under the constraints:
(i) n_4=1 or n_4=2
(ii) gcd(n, psi(n_2^2*n_3^3*n_4^4)) =1 where psi(k) = abs(A153038(k)) .

Examples

			1) Does not contain 10 = 10*1*1*1 where n_4=1 and gcd(10,|A153038(1)|)=1.
Both groups of order 10 have cyclic commutator groups: D10 has C5 and C10 has E.
2) Contains 12 = 3*2^2 where n_4=1 and gcd(12,|A153038(4)|) >1.
The group A4 of order 12 has a commutator group C2 x C2 which is not cyclic.
3) Contains 18 = 2*3^2 where n_4=1 and gcd(18,|A153038(9)|) >1.
The group (C3 x C3) : C2 of order 18 has a commutator group C3 x C3 which is not cyclic. (Gap notation, SmallGroup(18,4), where the colon is the semidirect product)
4) Contains 24 = 3*1*2^3 where n_4=1 and gcd(24,|A153038(9)|) >1.
5) Contains 32 = 1*1*1*1*2^5 where n_5>1.
6) Contains 48 = 3*1*1*2^4 where n_4=2 and gcd(48,|A153038(16)|)>1.

Links

Programs

  • Maple
    nsq := proc(n)
    local f,L ;
    L := [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1] ;
    if n = 1 then
        return L;
    else
        for f in ifactors(n)[2] do
            p := op(1,f) ;
            e := op(2,f) ;
            i := e ;
            L := subsop(i=op(i,L)*p^e,L) ;
        end do:
        return L ;
    end if;
end proc:
Pazdn4 := proc(L)
    if nops(L) <4 then
        1;
    else
        sqrt(sqrt(op(4,L))) ;
    end if;
end proc:
hihno1 := proc(L)
    i := 0 ;
    for j from 1 to nops(L) do
        if op(j,L) > 1 then
            i := j ;
        end if;
    end do:
    i ;
end proc:
for n from 1 to 600 do
    nf := nsq(n) ;
    n4 := Pazdn4(nf) ;
    psarg := op(2,nf)*op(3,nf)*op(4,nf) ;
    if  ( n4 =1 or n4 =2)  and gcd(n, abs(A153038(psarg))) = 1 and hihno1(nf) < 5 then
        ;
    else
        printf("%d,",n) ;
    end if;
end do:
  • Mathematica
    A153038[n_] := If[n == 1, 1, x = 1; Do[p = f[[1]]; e = f[[2]]; x = x*Product[1 - p^s, {s, 1, e}], {f, FactorInteger[n]}]; x];
nsq[n_] := Module[{f, L}, L = Table[1, {44}]; If[n == 1, Return[L], Do[{p, e} = f; L[[e]] = L[[e]]*p^e, {f, FactorInteger[n]}]]; L];
Pazdn4[L_] := If[Length[L] < 4, 1, Sqrt[Sqrt[L[[4]]]]];
hihno1[L_] := Module[{i, j}, i = 0; For[j = 1, j <= Length[L], j++, If[L[[j]] > 1, i = j]]; i];
Reap[For[n = 1, n <= 600, n++, nf = nsq[n]; n4 = Pazdn4[nf]; psarg = nf[[2]] * nf[[3]] * nf[[4]]; If[(n4 == 1 || n4 == 2) && GCD[n, Abs[A153038[psarg]]] == 1 && hihno1[nf] < 5, , Sow[n]]]][[2, 1]] (* Jean-François Alcover, Jun 07 2024, after R. J. Mathar *)
Showing 1-3 of 3 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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