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A290154 Smallest number k such that exactly half the numbers in [1..k] are prime(n)-smooth. 4
6, 20, 42, 78, 118, 184, 248, 332, 428, 534, 654, 772, 906, 1052, 1208, 1388, 1562, 1754, 1958, 2164, 2396, 2638, 2896, 3144, 3424, 3682, 3986, 4304, 4622, 4976, 5286, 5652, 6002, 6374, 6748, 7148, 7532, 7934, 8356, 8786, 9224, 9684, 10158, 10618, 11114, 11604 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

All terms are even numbers (because of the "exactly half the numbers in [1..k]" part of the definition).

LINKS

Robert Israel, Table of n, a(n) for n = 1..2000

EXAMPLE

The 2-smooth numbers are 1, 2, 4, 8, 16, 32, ... (A000079, the powers of 2), so the numbers of 2-smooth numbers in the interval [1..k] for k = 2, 4, and 6 are 2, 3, and 3, respectively; thus, the smallest k at which the number of 2-smooth numbers in [1..k] is exactly k/2 is k=6, so a(1)=6.

The 3-smooth numbers are 1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 27, 32, ... (A003586), so there are more than k/2 3-smooth numbers in [1..k] for every positive k < 20, but exactly k/2 3-smooth numbers in [1..20], so a(2) = 20.

MAPLE

N:= 100:

mypi:= proc(n) option remember; global pmax; local k;

  k:= procname(pmax);

  while pmax < n do pmax:= nextprime(pmax); k:= k+1 od;

  k

end proc:

pmax:= 2: mypi(2):= 1:

V:= Vector(N):

count:= 0:

loheap:=heap[new](`<`, 0): nlo:= 1:

hiheap:= heap[new](`>`, 1): nhi:= 1:

for k from 4 by 2 while count < N do

   for v in [mypi(max(numtheory:-factorset(k-1))), mypi(max(numtheory:-factorset(k)))] do

     if v <= heap[max](loheap) then heap[insert](v, loheap); nlo:= nlo+1;

     elif v >= heap[max](hiheap) then heap[insert](v, hiheap); nhi:= nhi+1;

     elif nlo <= nhi then heap[insert](v, loheap); nlo:= nlo+1;

     else heap[insert](v, hiheap); nhi:= nhi+1;

     fi;

   od;

   if nlo < nhi-1 then

        t:= heap[extract](hiheap);

        heap[insert](t, loheap);

        nlo:= nlo+1; nhi:= nhi-1;

   elif nhi < nlo-1 then

        t:= heap[extract](loheap);

        heap[insert](t, hiheap);

        nhi:= nhi+1; nlo:= nlo-1;

   fi;

   for n from heap[max](loheap) to min(heap[max](hiheap)-1, N) do

     if V[n] = 0 then count:= count+1; V[n]:= k;

     fi

   od;

od:

convert(V, list); # Robert Israel, Mar 28 2019

MATHEMATICA

smoothQ[k_, p_] := k <= p || Max[FactorInteger[k][[All, 1]]] <= p; a[n_] := For[p = Prime[n]; cnt = 0; k = 1, True, k++, If[smoothQ[k, p], cnt++]; If[cnt == k/2, Return[k]]]; Array[a, 46] (* Jean-Fran├žois Alcover, Jul 22 2017 *)

PROG

(PARI) is(k, n) = {m=k; forprime(p=2, prime(n), while(m%p==0, m=m/p)); return(m==1); }

a(n) = {j=2; x=2; y=0; while(x!=y, j+=2; s=is(j, n)+is(j-1, n); x+=s; y+=2-s); j; } \\ Jinyuan Wang, Aug 03 2019

CROSSREFS

Cf. A000079, A003586, A126283.

Sequence in context: A002943 A068377 A009946 * A094274 A094279 A093913

Adjacent sequences:  A290151 A290152 A290153 * A290155 A290156 A290157

KEYWORD

nonn

AUTHOR

Jon E. Schoenfield, Jul 21 2017

STATUS

approved

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Last modified October 17 22:05 EDT 2019. Contains 328134 sequences. (Running on oeis4.)