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
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
KEYWORD
nonn
AUTHOR
Robert G. Wilson v, Oct 12 2017
EXTENSIONS
More terms from Robert Israel, Nov 01 2017
STATUS
approved