This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A384195 #14 May 27 2025 19:09:12 %S A384195 1,1,0,1,0,0,1,0,-1,2,0,-2,2,1,-2,1,0,0,2,-2,-2,4,1,-4,1,2,-2,2,0,-2, %T A384195 2,-2,0,5,-2,-5,2,4,-2,0,0,-2,4,-2,-4,6,1,-4,1,0,-2,2,2,0,0,-4,-2,8,0, %U A384195 -8,4,3,-2,1,-2,-2,2,2,-2,4,0,-8,4,2,-2,2,-2,2,3,-6,-3,8,2,-8 %N A384195 a(n) = tau(n+1) - tau(n-1), where tau(n) = A000005(n), the number of divisors of n. %C A384195 Conjecture: a(n)=0 for an infinite number of values. %F A384195 a(n) = tau(n+1) - tau(n-1). %e A384195 a(10) = tau(11) - tau(9) = 2 - 3 = -1. %o A384195 (Haskell) %o A384195 divides ∷ (Integral a) ⇒ a → a → Bool %o A384195 divides a b = b `mod` a == 0 %o A384195 properFactors ∷ Integral a ⇒ a → [a] %o A384195 properFactors n = 1 : filter (`divides` n) [2..(n `div` 2)] %o A384195 factors ∷ Integral a ⇒ a → [a] %o A384195 factors 1 = [1] %o A384195 factors n = properFactors n <> [n] %o A384195 tauSuccMinusTauPred :: Integer -> Integer %o A384195 tauSuccMinusTauPred n = fromIntegral (length (factors (n + 1))) - fromIntegral (length (factors (n - 1))) %o A384195 (PARI) a(n) = numdiv(n+1) - numdiv(n-1); \\ _Michel Marcus_, May 21 2025 %Y A384195 Cf. A000005, A051950, A051951, A067888. %K A384195 sign %O A384195 2,10 %A A384195 _Dan Dart_, May 21 2025