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

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, 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, 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

Offset: 1

Author

Vladimir Joseph Stephan Orlovsky, Apr 10 2010

Keywords

• nonn
• easy

Comments

From Amiram Eldar, Mar 28 2025: (Start)

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.

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.

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)

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000217, A051903, A135939, A334572.

Programs

Mathematica

f1[n_] := Max[Last/@FactorInteger[n]]; f2[n_]:=n*(n+1)/2; Join[{0}, Table[f1[f2[n]], {n, 2, 4*5!}]]
Max[FactorInteger[#][[;;,2]]]&/@Accumulate[Range[110]] // ReplacePart[1 -> 0] (* Harvey P. Dale, Oct 23 2024 *)

PARI

a(n) = if(n == 1, 0, vecmax(factor(n*(n+1)/2)[, 2])); \\ Amiram Eldar, Mar 28 2025

Formula

From Amiram Eldar, Mar 28 2025: (Start)

a(n) = A051903(A000217(n)).

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)

Extensions

a(1) inserted by Amiram Eldar, Mar 28 2025