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

Author

Luca Pezzullo, Jun 11 2016

Keywords

• nonn

Examples

			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.
		

Links

• Eric Weisstein's World of Mathematics, Sum of prime factors
• Eric Weisstein's World of Mathematics, Triangular number

Crossrefs

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

Programs

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 *)

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

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)).

Extensions

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