A005236 Barriers for omega(n): numbers n such that, for all m < n, m + omega(m) <= n. (Formerly M0501)

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

1,1

Comments

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

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.

Example

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.

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 *)

Prog

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

Crossrefs

Cf. A001221, A229109.

Sequence in context: A164043 A172248 A082415 * A051250 A143071 A305759

Adjacent sequences: A005233 A005234 A005235 * A005237 A005238 A005239

Keyword

nonn,nice

Author

N. J. A. Sloane

Extensions

More terms from John W. Layman

Status

approved