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A066728 a(n) is the number of integers of the form (n+k+n*k)/(n-k) for k = 1,2,...,n-1.

%I A066728 #16 Feb 01 2025 08:44:24
%S A066728 0,1,1,3,1,4,2,4,2,7,1,7,3,5,3,8,1,11,3,7,3,9,2,9,5,7,3,15,1,13,3,6,7,
%T A066728 11,3,11,3,9,3,19,1,15,5,7,5,11,2,17,5,11,3,15,3,19,7,9,3,15,1,15,5,7,
%U A066728 11,15,3,15,3,15,3,29,1,14,5,7,11,15,3,23,4,11,4,15,3,15,7,9,3,29,3,23
%N A066728 a(n) is the number of integers of the form (n+k+n*k)/(n-k) for k = 1,2,...,n-1.
%C A066728 a(n) = 1 iff n is 2 or the lesser of twin primes (for n >= 3, n follows the sequence A001359).
%C A066728 Also the number of factors of n*(n+2) which are less than n. - Antonio G. Astudillo (afg_astudillo(AT)lycos.com), Jul 02 2003
%H A066728 Amiram Eldar, <a href="/A066728/b066728.txt">Table of n, a(n) for n = 1..10000</a>
%F A066728 a(n) = ceiling( d(n*(n+2)) / 2 ) - 1, where d(n) = number of divisors of n (A000005). - Antonio G. Astudillo (afg_astudillo(AT)lycos.com), Jul 02 2003
%e A066728 (4 + 1 + 4*1)/(4 - 1), (4 + 2 + 4*2)/(4 - 2), and (4 + 3 + 4*3)/(4 - 1) are integers, hence a(4) = 3.
%p A066728 with(numtheory):A066728 := n->ceil(tau(n*(n+2))/2)-1;
%t A066728 a[n_] := Ceiling[DivisorSigma[0, n*(n+2)] / 2]  - 1; Array[a, 100] (* _Amiram Eldar_, Feb 01 2025 *)
%o A066728 (PARI) a(n) = ceil(numdiv(n*(n+2))/2) - 1; \\ _Amiram Eldar_, Feb 01 2025
%Y A066728 Cf. A000005, A001359, A063091.
%K A066728 nonn,easy
%O A066728 1,4
%A A066728 _Benoit Cloitre_, Jan 15 2002