A005236 - OEIS

%I M0501 #42 Sep 07 2023 04:45:11

%S 2,3,4,5,6,8,9,10,12,14,17,18,20,24,26,28,30,33,38,42,48,50,54,60,65,

%T 74,82,84,90,98,102,108,110,114,126,129,138,150,164,168,174,180,194,

%U 198,228,234,244,252,258,264,270,290,294,318,348,354,360,384,390,402

%N Barriers for omega(n): numbers n such that, for all m < n, m + omega(m) <= n.

%C omega(m) is the number of distinct prime factors of m.

%D R. K. Guy, Unsolved Problems in Number Theory, B8.

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%H Charles R Greathouse IV, <a href="/A005236/b005236.txt">Table of n, a(n) for n = 1..10000</a>

%H Paul Erdős, <a href="https://www.jstor.org/stable/2689842">Some Unconventional Problems in Number Theory</a>, Mathematics Magazine, Vol. 52, No. 2, Mar., 1979, pp. 67-70. See Problem 4, p. 68.

%H Paul Erdős, <a href="https://users.renyi.hu/~p_erdos/1979-23.pdf">Some unconventional problems in number theory</a>, Acta Mathematica Hungarica, 33(1):71-80, 1979.

%e 1 + omega(1) = 1, 2 + omega(2) = 3, 3 + omega(3) = 4, 4 + omega(4) = 5, 5 + omega(5) = 6.

%e Thus we have verified that m + omega(m) < 6 for m < 6, so 6 is in the sequence.

%e But since 6 + omega(6) = 8 > 7, 7 is not in the sequence.

%t omegaBarrierQ[n_] := (For[m = 1, m < n, m++, If[m + PrimeNu[m] > n, Return[False]]]; True); Select[Range[2, 500], omegaBarrierQ] (* _Jean-François Alcover_, Feb 03 2015 *)

%o (PARI) is(n)=for(k=1,log(n)\log(5),if(omega(n-k)>k,return(0)));n>1 \\ _Charles R Greathouse IV_, Sep 19 2012

%o (Haskell)

%o a005236 n = a005236_list !! (n-1)

%o a005236_list = filter (\x -> all (<= x) $ map a229109 [1..x-1]) [2..]

%o -- _Reinhard Zumkeller_, Sep 13 2013

%Y Cf. A001221, A229109.

%K nonn,nice

%O 1,1

%A _N. J. A. Sloane_

%E More terms from _John W. Layman_