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

1,2

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

Formula

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

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 *)

Crossrefs

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

Sequence in context: A100637 A081063 A081304 * A245339 A324161 A028904

Adjacent sequences: A137718 A137719 A137720 * A137722 A137723 A137724

Keyword

nonn

Author

Reinhard Zumkeller, Feb 09 2008

Status

approved