A005236 Barriers for omega(n): numbers n such that, for all m < n, m + omega(m) <= n.

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, 74, 82, 84, 90, 98, 102, 108, 110, 114, 126, 129, 138, 150, 164, 168, 174, 180, 194, 198, 228, 234, 244, 252, 258, 264, 270, 290, 294, 318, 348, 354, 360, 384, 390, 402

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• nice

Comments

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

Examples

			1 + omega(1) = 1, 2 + omega(2) = 3, 3 + omega(3) = 4, 4 + omega(4) = 5, 5 + omega(5) = 6.
Thus we have verified that m + omega(m) < 6 for m < 6, so 6 is in the sequence.
But since 6 + omega(6) = 8 > 7, 7 is not in the sequence.
		

References

• R. K. Guy, Unsolved Problems in Number Theory, B8.
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
• Paul Erdős, Some Unconventional Problems in Number Theory, Mathematics Magazine, Vol. 52, No. 2, Mar., 1979, pp. 67-70. See Problem 4, p. 68.
• Paul Erdős, Some unconventional problems in number theory, Acta Mathematica Hungarica, 33(1):71-80, 1979.

Crossrefs

Cf. A001221, A229109.

Programs

Haskell

a005236 n = a005236_list !! (n-1)
a005236_list = filter (\x -> all (<= x) $ map a229109 [1..x-1]) [2..]
-- Reinhard Zumkeller, Sep 13 2013

Mathematica

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 *)

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

Extensions

More terms from John W. Layman