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A176166 a(n) is the maximum exponent in the prime factorization of the n-th triangular number.

%I A176166 #14 Mar 28 2025 04:18:05
%S A176166 0,1,1,1,1,1,2,2,2,1,1,1,1,1,3,3,2,2,1,1,1,1,2,2,2,3,3,1,1,1,4,4,1,1,
%T A176166 2,2,1,1,2,2,1,1,1,2,2,1,3,3,2,2,1,1,3,3,2,2,1,1,1,1,1,2,5,5,1,1,1,1,
%U A176166 1,1,2,2,1,2,2,1,1,1,3,4,4,1,1,1,1,1,2,2,2,2,1,1,1,1,4,4,2,2,2,2,1,1,2,2,1,1
%N A176166 a(n) is the maximum exponent in the prime factorization of the n-th triangular number.
%C A176166 From _Amiram Eldar_, Mar 28 2025: (Start)
%C A176166 The asymptotic density of the occurrences of terms < k is f(k) = Product_{p prime} (1 - 2/p^k) * (1+1/(2*(2^(k-1)-1))), for k >= 2.
%C A176166 The asymptotic density of the occurrences of k is d(1) = f(2) for k = 1, and d(k) = f(k+1) - f(k) for k >= 2.
%C A176166 The asymptotic mean of this sequence is lim_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Sum_{k>=1} k*d(k) = 1 + Sum_{k>=1} (1 - Product_{p prime} (1 - 2/p^(k+1)) * (1+1/(2*(2^k-1)))) = 1.89137712344735606085... . (End)
%H A176166 Amiram Eldar, <a href="/A176166/b176166.txt">Table of n, a(n) for n = 1..10000</a>
%F A176166 From _Amiram Eldar_, Mar 28 2025: (Start)
%F A176166 a(n) = A051903(A000217(n)).
%F A176166 a(n) = max(A051903(n), A051903((n+1)/2)) if n is odd, and max(A051903(n/2), A051903(n+1)) if n is even. (End)
%t A176166 f1[n_] := Max[Last/@FactorInteger[n]]; f2[n_]:=n*(n+1)/2; Join[{0}, Table[f1[f2[n]], {n, 2, 4*5!}]]
%t A176166 Max[FactorInteger[#][[;;,2]]]&/@Accumulate[Range[110]] // ReplacePart[1 -> 0] (* _Harvey P. Dale_, Oct 23 2024 *)
%o A176166 (PARI) a(n) = if(n == 1, 0, vecmax(factor(n*(n+1)/2)[, 2])); \\ _Amiram Eldar_, Mar 28 2025
%Y A176166 Cf. A000217, A051903, A135939, A334572.
%K A176166 nonn,easy
%O A176166 1,7
%A A176166 _Vladimir Joseph Stephan Orlovsky_, Apr 10 2010
%E A176166 a(1) inserted by _Amiram Eldar_, Mar 28 2025