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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

%I #39 Mar 12 2024 02:41:40

%S 4,1,2,6,12,24,48,60,192,120,180,240,3072,360,12288,960,720,840,

%T 196608,1260,786432,1680,2880,15360,12582912,2520,6480,61440,6300,

%U 6720,805306368,5040,3221225472,7560,46080,983040,25920,10080,206158430208,3932160,184320,15120,3298534883328,20160,13194139533312,107520,25200,62914560

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

%C First occurrence of k in A292286.

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

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

%C From _Robert Israel_, Nov 01 2017: (Start)

%C All terms are in A025487.

%C 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).

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

%C (End)

%H Robert Israel, <a href="/A293570/b293570.txt">Table of n, a(n) for n = -1..3318</a>

%H Robert G. Wilson v, <a href="/A293570/a293570.txt">Table of n and a(n) for n = -1..500, or 0 if no such value is known plus other known values.</a>

%H Robert Israel, <a href="/A293570/a293570_1.txt">Table of n and a(n) for all terms where a(n) <= 201906284580000</a>

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

%p g:= proc(F, k)

%p # minimize Product_{i>=k} prime(i)^(e(i)-1) s.t. Product_{i>=k} e(i) = n

%p # return [v, E] where E the list of e(i) and v the value

%p # F the prime factorization of n

%p uses combinat;

%p local e, pk, Fv, gv, v, vmin, gmin, T, t, gpf;

%p if F = [] then return [1,[]] fi;

%p vmin:= infinity;

%p gpf:= F[-1][1];

%p pk:= ithprime(k);

%p T:= cartprod([seq([$0..f[2]], f = F)]);

%p while not T[finished] do

%p t:= T[nextvalue]();

%p e:= mul(F[i][1]^t[i], i=1..nops(F));

%p if e < gpf then next fi;

%p Fv:= [seq(`if`(t[i] = F[i][2], NULL, [F[i][1],F[i][2]-t[i]]),i=1..nops(F))];

%p gv:= procname(Fv,k+1);

%p v:= pk^(e-1) * gv[1];

%p if v < vmin then

%p vmin:= v;

%p gmin:= [e,op(gv[2])];

%p fi

%p od;

%p [vmin, gmin]

%p end proc:

%p 4,seq(g(ifactors(2*n)[2],1)[1], n=0..50); # _Robert Israel_, Nov 01 2017

%t 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]

%o (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

%Y Cf. A003680, A007955, A025487, A292286.

%K nonn

%O -1,1

%A _Robert G. Wilson v_, Oct 12 2017

%E More terms from _Robert Israel_, Nov 01 2017

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Last modified March 29 00:26 EDT 2024. Contains 371264 sequences. (Running on oeis4.)