A211872 For each triprime (A014612) less than or equal to n, sum the positive integers less than or equal to the number of divisors of the triprime.
0, 0, 0, 0, 0, 0, 0, 10, 10, 10, 10, 31, 31, 31, 31, 31, 31, 52, 52, 73, 73, 73, 73, 73, 73, 73, 83, 104, 104, 140, 140, 140, 140, 140, 140, 140, 140, 140, 140, 140, 140, 176, 176, 197, 218, 218, 218, 218, 218, 239, 239, 260, 260, 260, 260, 260, 260, 260, 260
Offset: 1
Keywords
Examples
a(1) = a(2) = a(3) = a(4) = a(5) = a(6) = a(7) = 0. a(8) = 10 since 8 has 4 divisors, and the sum of all the numbers up to 4 is 1 + 2 + 3 + 4 = 10. The next triprime is 12, so a(8) = a(9) = a(10) = a(11) = 10. Since there are two triprimes less than or equal to 12, we sum the numbers from 1 to d(8) and 1 to d(12), then take the sum total. Thus, a(12) = 10 + 21 = 31.
Links
- G. C. Greubel, Table of n, a(n) for n = 1..5000
Programs
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Mathematica
sm = 0; Table[If[Total[Transpose[FactorInteger[n]][[2]]] == 3, d = DivisorSigma[0, n]; sm = sm + d (d + 1)/2]; sm, {n, 100}] (* T. D. Noe, Feb 14 2013 *) Table[Sum[KroneckerDelta[PrimeOmega[i], 3]*Sum[j, {j, DivisorSigma[0, i]}], {i, n}], {n, 50}] (* Wesley Ivan Hurt, Oct 07 2014 *)
Formula
a(n) = Sum_{i=1..n} [Omega(i) = 3] * Sum_{j = 1..d(i)} j.
a(n) = Sum_{i=1..n} [Omega(i) = 3] * (omega(i) + 1) * (d(i) + 1).
a(n) = Sum_{i=1..n} [Omega(i) = 3] * (2*omega(i)^2 + 5*omega(i) + 3), where [ ] is the Iverson bracket.
Comments