A293570 a(-1)=4; thereafter a(n) is the least integer m such that the product of the divisors of m is m^n.

4, 1, 2, 6, 12, 24, 48, 60, 192, 120, 180, 240, 3072, 360, 12288, 960, 720, 840, 196608, 1260, 786432, 1680, 2880, 15360, 12582912, 2520, 6480, 61440, 6300, 6720, 805306368, 5040, 3221225472, 7560, 46080, 983040, 25920, 10080, 206158430208, 3932160, 184320, 15120, 3298534883328, 20160, 13194139533312, 107520, 25200, 62914560

Offset

-1,1

Comments

First occurrence of k in A292286.

Records occur for 4, 6, 12, 24, 48, 60, 192, 240, 3072, 12288, 196608, 786432, 12582912, 805306368, 3221225472, etc.

Terms not a multiple of 60: 1, 2, 4, 6, 12, 24, 48, 192, 3072, 12288, 196608, 786432, 12582912, 805306368, 3221225472, etc.

From Robert Israel, Nov 01 2017: (Start)

All terms are in A025487.

For n >= 1, if a(n) = Product_{i=1..k} prime(i)^e(i) then n = (1/2)*Product_{i=1..k} (e(i)+1).

If p is prime, a(p) = 2^(p-1)*3.

(End)

Links

Robert Israel, Table of n, a(n) for n = -1..3318

Robert G. Wilson v, Table of n and a(n) for n = -1..500, or 0 if no such value is known plus other known values.

Robert Israel, Table of n and a(n) for all terms where a(n) <= 201906284580000

Formula

a(n) = A003680(n), for n >= 1.

Maple

g:= proc(F, k)
# minimize Product_{i>=k} prime(i)^(e(i)-1) s.t. Product_{i>=k} e(i) = n
# return [v, E] where E the list of e(i) and v the value
# F the prime factorization of n
  uses combinat;
  local e, pk, Fv, gv, v, vmin, gmin, T, t, gpf;
  if F = [] then return [1, []] fi;
  vmin:= infinity;
  gpf:= F[-1][1];
  pk:= ithprime(k);
  T:= cartprod([seq([$0..f[2]], f = F)]);
  while not T[finished] do
    t:= T[nextvalue]();
    e:= mul(F[i][1]^t[i], i=1..nops(F));
    if e < gpf then next fi;
    Fv:= [seq(`if`(t[i] = F[i][2], NULL, [F[i][1], F[i][2]-t[i]]), i=1..nops(F))];
    gv:= procname(Fv, k+1);
    v:= pk^(e-1) * gv[1];
    if v < vmin then
       vmin:= v;
       gmin:= [e, op(gv[2])];
    fi
  od;
  [vmin, gmin]
end proc:
4, seq(g(ifactors(2*n)[2], 1)[1], n=0..50); # Robert Israel, Nov 01 2017

Mathematica

f[n_] := Boole[n == 1] + If[OddQ@#, -1, #/2] &@DivisorSigma[0, n]; t[_] = 0; k = 1; While[k < 3300000000, a = f@k; If[ t[a] == 0, t[a] = k; Print[{a, k}]]; k ++]; t@# & /@ Range[-1, 36]

Prog

(PARI) a(n) = if(n == 0 || n == -1, return((n-1)^2)); for(m=2, +oo, my(p=1); fordiv(m, d, p*=d); if(p == m^n, return(m))) \\ Iain Fox, Dec 14 2017

Crossrefs

Cf. A003680, A007955, A025487, A292286.

Sequence in context: A343404 A220182 A076064 * A016685 A141332 A241184

Adjacent sequences: A293567 A293568 A293569 * A293571 A293572 A293573

Keyword

nonn

Author

Robert G. Wilson v, Oct 12 2017

Extensions

More terms from Robert Israel, Nov 01 2017

Status

approved