

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.


1



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
(list;
graph;
refs;
listen;
history;
text;
internal format)



OFFSET

1,8


COMMENTS

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.


LINKS

G. C. Greubel, Table of n, a(n) for n = 1..5000


FORMULA

a(n) = sum(i <= n and Omega(i) = 3, sum(j = 1..d(i), j ) ).
a(n) = sum(i <= n and Omega(i) = 3, ( omega(i) + 1 ) * ( d(i) + 1 ) ).
a(n) = sum(i <= n and Omega(i) = 3, (1/2)*d(i)^2  (1/2)*d(i)  1) ).
a(n) = sum(i <= n and Omega(i) = 3, 2*omega(i)^2 + 5*omega(i) + 3 ).


EXAMPLE

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.


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 *)


CROSSREFS

Cf. A000005, A001221, A001222, A014612, A209323.
Sequence in context: A316650 A216875 A166710 * A178166 A003855 A245431
Adjacent sequences: A211869 A211870 A211871 * A211873 A211874 A211875


KEYWORD

nonn


AUTHOR

Wesley Ivan Hurt, Feb 12 2013


STATUS

approved



