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A293570 a(-1)=4; thereafter a(n) is the least integer m such that the product of the divisors of m is m^n. 2
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 (list; graph; refs; listen; history; text; internal format)
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 > 0. - Paolo P. Lava, May 04 2018

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: A087225 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

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Last modified October 20 05:49 EDT 2019. Contains 328247 sequences. (Running on oeis4.)