A274140 Sum of primes dividing n-th triangular number, counted with multiplicity.

0, 3, 5, 7, 8, 10, 11, 10, 11, 16, 16, 18, 20, 15, 14, 23, 23, 25, 26, 17, 21, 34, 30, 17, 23, 22, 18, 38, 37, 39, 39, 22, 31, 29, 20, 45, 56, 35, 25, 50, 51, 53, 56, 24, 34, 70, 56, 23, 24, 30, 35, 68, 62, 25, 27, 33, 51, 88, 69, 71, 92, 44, 23, 28, 32, 81, 86, 45, 38, 83, 81, 83, 110, 50, 34, 39, 34, 95, 90

Offset

1,2

Links

Table of n, a(n) for n=1..79.

Eric Weisstein's World of Mathematics, Sum of prime factors

Eric Weisstein's World of Mathematics, Triangular number

Formula

For any integer coefficient C(n) of the polynomial generated by the Triangular Numbers generating function f(x)=x/((1-x)^3), if C(n) = Product (p_j^k_j) then a(n) = Sum (p_j * k_j).

a(n) = A001414(A000217(n)).

Example

a(4) = 7; the 4th triangular number is 10, the prime factors of 10 are 2 and 5, and 2+5 = 7.
a(6) = 10; the 6th triangular number is 21, the prime factors of 21 are 3 and 7, and 3+7 = 10.

Mathematica

a[1]=0; a[n_] := Plus @@ Times @@@ FactorInteger[n (n+1)/2]; Array[a, 80] (* Giovanni Resta, Jun 12 2016 *)
Join[{0}, Rest[Total[Times@@@FactorInteger[#]]&/@Accumulate[Range[100]]]] (* Harvey P. Dale, May 06 2024 *)

Prog

(PARI) a(n) = my(f=factor(n*(n+1)/2)); sum(i=1, matsize(f)[1], f[i, 1]*f[i, 2]) \\ David A. Corneth, Jun 12 2016

Crossrefs

Cf. A000217 (triangular numbers), A001414 (sum of primes dividing n).

Sequence in context: A175144 A183054 A188569 * A212294 A299495 A186689

Adjacent sequences: A274137 A274138 A274139 * A274141 A274142 A274143

Keyword

nonn

Author

Luca Pezzullo, Jun 11 2016

Extensions

a(30) and a(38) corrected by Giovanni Resta, Jun 12 2016

Status

approved