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A211872
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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.
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1
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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
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OFFSET
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1,8
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COMMENTS
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The largest difference between any pair of consecutive numbers in the sequence = 36, The second largest difference = 21, the third largest = 10, and the fourth (and last) possible difference is 0.
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LINKS
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FORMULA
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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.
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EXAMPLE
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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.
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MATHEMATICA
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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 *)
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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