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 A129979 #11 Jul 20 2025 06:00:24 %S A129979 1,3,3,2,3,1,3,2,2,1,3,2,3,1,1,2,3,2,3,2,1,1,3,2,2,1,2,2,3,3,3,2,1,1, %T A129979 1,2,3,1,1,2,3,3,3,2,2,1,3,2,2,2,1,2,3,2,1,2,1,1,3,2,3,1,2,2,1,3,3,2, %U A129979 1,3,3,2,3,1,2,2,1,3,3,2,2,1,3,2,1,1,1,2,3,2 %N A129979 a(n) = 2-mu(n), where mu=A008683 is the Moebius function. %C A129979 Left border of the triangle A131088. %C A129979 From _Wesley Ivan Hurt_, Aug 22 2013 (Start): %C A129979 1 <= a(n) <= 3: a(n) = 1 when n is both squarefree and has an even number of distinct prime factors (or if n = 1). So a(n) = 1 when mu(n) = 1. a(n) = 2 when n is square-full. a(n) = 3 when n is both squarefree and has an odd number of distinct prime factors. %C A129979 When n is semiprime, a(n) is equal to the ratio of the number of prime factors of n (with multiplicity) to the number of its distinct prime factors. Analogously, when n is semiprime, a(n) is equal to the ratio of the sum of the prime factors of n (with repetition) to the sum of its distinct prime factors. %C A129979 (End) %F A129979 Inverse Moebius transform of A007427 with changed signs except for A007427(1) = 1; i.e., inverse Moebius transform of (1, 2, 2, -1, 2, -4, 2, 0, -1, -4, ...). %F A129979 a(n) = 2 - mu(n) = 2 - A008683(n). - _Wesley Ivan Hurt_, Aug 22 2013 %F A129979 a(A001358(n)) = 5 - tau(A001358(n)) = 3 - omega(A001358(n)) = 3 + 2*A001358(n) - sigma(A001358(n)) - phi(A001358(n)) = Omega(A001358(n))/omega(A001358(n))= A001414(A001358(n))/A008472(A001358(n)). - _Wesley Ivan Hurt_, Aug 22 2013 %e A129979 A131088 = (1; 3,1; 3,0,1; 2,3,0,1; ...). %p A129979 with(numtheory); seq(2-mobius(k),k=1..70); # _Wesley Ivan Hurt_, Aug 22 2013 %t A129979 2 - MoebiusMu[Range[100]] (* _Alonso del Arte_, Aug 22 2013 *) %o A129979 (PARI) T(n,k) = 2*!(n%k) - if (!(n % k), moebius(n/k), 0); \\ A131088 %o A129979 a(n) = T(n, 1); \\ _Michel Marcus_, Feb 26 2022 %Y A129979 Cf. A131088, A051731, A054525, A007427. %K A129979 nonn,easy %O A129979 1,2 %A A129979 _Gary W. Adamson_, Jun 14 2007 %E A129979 More terms from _Michel Marcus_, Feb 26 2022