A137721 Number of numbers not greater than n with no prime gaps in their factorization.

1, 2, 3, 4, 5, 6, 7, 8, 9, 9, 10, 11, 12, 12, 13, 14, 15, 16, 17, 17, 17, 17, 18, 19, 20, 20, 21, 21, 22, 23, 24, 25, 25, 25, 26, 27, 28, 28, 28, 28, 29, 29, 30, 30, 31, 31, 32, 33, 34, 34, 34, 34, 35, 36, 36, 36, 36, 36, 37, 38, 39, 39, 39, 40, 40, 40, 41, 41, 41, 41, 42, 43, 44

Offset: 1

Author

Reinhard Zumkeller, Feb 09 2008

Keywords

• nonn

Comments

a(n) > a(n-1) iff A073490(n) = 0;

a(n) > A137722(n) for n < 134;

a(n) < A137722(n) for n > 140;

a(A137723(n) + n) = a(A137723(n)) + 1.

Partial sums of A137794. - Reinhard Zumkeller, Feb 11 2008

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A073490, A073491, A137722, A137723, A137794.

Programs

Mathematica

b[n_] := With[{pp = PrimePi @ FactorInteger[ n ][[All, 1]]},
     Boole[pp[[-1]] - pp[[1]] + 1 == Length[pp]]]; (* b is A137794 *)
Array[b, 105] // Accumulate (* Jean-François Alcover, Dec 09 2021 *)

Formula

a(n) = Sum_{k=1..n} 0^A073490(k).