A380744 Number of integers strictly between n^2 and (n+1)^2 with at most four prime factors (counting multiplicity).

2, 4, 6, 8, 9, 11, 14, 14, 17, 17, 21, 21, 23, 24, 27, 27, 30, 33, 33, 34, 36, 37, 41, 40, 43, 43, 46, 49, 51, 50, 52, 53, 54, 58, 60, 61, 61, 61, 66, 66, 65, 73, 72, 75, 74, 75, 79, 79, 79, 80, 83, 88, 87, 90, 90, 92, 93, 93, 98, 97, 98, 99, 104, 105, 102

Offset

1,1

Links

Table of n, a(n) for n=1..65.

Viggo Brun, Le crible d'Eratosthène et le théorème de Goldbach, Skrifter utgitt av det Norske Videnskaps-Akademi, Matematisk-Naturvidenskapelig Klasse Kristiana 3 (1920).

Adrian W. Dudek and Daniel R. Johnston, Almost primes between all squares, arXiv preprint arXiv:2501.18048 [math.NT], 2025.

Formula

Dudek & Johnston prove that a(n) > 0 for all n.

Mathematica

a[n_]:=Length[Select[Range[n^2+1, n(n+2)], PrimeOmega[#]<5 &]]; Array[a, 65] (* Stefano Spezia, Feb 01 2025 *)

Prog

(PARI) a(n)=my(s); forfactored(k=n^2+1, n^2+2*n, bigomega(k)<5 && s++); s

Crossrefs

Cf. A001222, A014085.

Sequence in context: A184587 A345436 A225773 * A233671 A348854 A143346

Adjacent sequences: A380741 A380742 A380743 * A380745 A380746 A380747

Keyword

nonn

Author

Charles R Greathouse IV, Jan 31 2025

Status

approved