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

1,7

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

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)

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 *)

Prog

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

Crossrefs

Cf. A000217, A051903, A135939, A334572.

Sequence in context: A126061 A088496 A036602 * A167911 A037804 A396299

Adjacent sequences: A176163 A176164 A176165 * A176167 A176168 A176169

Keyword

nonn,easy

Author

Vladimir Joseph Stephan Orlovsky, Apr 10 2010

Extensions

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

Status

approved