A370935 - OEIS

%I #60 May 10 2024 11:08:46

%S 1,0,9,8,189,72,160,128,384,1152,3456,6912,10240,46080,55296,32768,

%T 98304,294912,884736,4423680,3538944,13107200,27262976,73400320,

%U 41943040,254803968,226492416,1132462080,2038431744,1811939328,9059696640,2147483648,6442450944,19327352832,57982058496,289910292480

%N a(n) is the first term of A351048 with n prime factors, counted with multiplicity, or 0 if there is no such term.

%C a(n) is the first number k, if any, such that A001222(k) = n and A000005(k) divides A003415(k).

%H Robert Israel, <a href="/A370935/b370935.txt">Table of n, a(n) for n = 0..1000</a>

%F a(2^k - 1) = 2^(2^k - 1) for k >= 2.

%e a(4) = 189 because 189 = 3^3 * 7 has 4 prime factors and A000005(189) = 8 divides A003415(189) = 216, and no smaller number works.

%p children:= proc(r) local L, x, p, q, t, R;

%p   x:= r[1];

%p   L:= r[2];

%p   t:= L[-1];

%p   p:= t[1]; q:= nextprime(p);

%p   if t[2]=1 then t:= [q, 1];

%p   else t:= [p, t[2]-1], [q, 1]

%p   fi;

%p   R:= [x*q/p, [op(L[1..-2]), t]];

%p   if nops(L) >= 2 then

%p     p:= L[-2][1];

%p     q:= L[-1][1];

%p     if L[-2][2]=1 then t:= [q, L[-1][2]+1]

%p     else t:= [p, L[-2][2]-1], [q, L[-1][2]+1]

%p     fi;

%p     R:= R, [x*q/p, [op(L[1..-3]), t]]

%p   fi;

%p   [R]

%p end proc:

%p f:= proc(n)

%p uses priqueue;

%p local pq, t, x, V,F;

%p initialize(pq);

%p insert([-2^n, [[2,n]]],pq);

%p do

%p   V:= extract(pq);

%p   x:= -V[1]; F:= V[2];

%p   if (x * add(t[2]/t[1], t=F)) mod mul(t[2]+1, t=F) = 0

%p      then return(x) fi;

%p   for t in children(V) do insert(t,pq) od;

%p od;

%p end proc:

%p 1, 0, seq(f(n),n=2..40);

%Y Cf. A000005, A001222, A003415, A351048.

%K nonn

%O 0,3

%A _Robert Israel_, May 06 2024