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, Math. Mag. (March 1979) Vol. 52, No. 2, 67-70. See Problem 4, p. 68.
Paul Erdős, Some unconventional problems in number theory, Acta Math. Hungarica (1979) Vol. 33, No. 1, 71-80.
Terence Tao and Joni Teräväinen, Quantitative correlations and some problems on prime factors of consecutive integers, arXiv:2512.01739 [math.NT], 2025. See p. 3.
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
KEYWORD
nonn,nice
AUTHOR
EXTENSIONS
More terms from John W. Layman
STATUS
approved