A090498 Number of divisors of all the numbers from (1/2)n(n-1)+1 to n(n+1)/2, i.e., tau(1), tau(2)+tau(3), tau(4)+tau(5)+tau(6), tau(7)+tau(8)+tau(9)+tau(10), ..., where tau(j) is the number of divisors of j.

1, 4, 9, 13, 18, 25, 31, 39, 42, 49, 61, 64, 73, 81, 92, 93, 101, 115, 120, 135, 131, 148, 157, 165, 171, 178, 195, 195, 210, 219, 229, 238, 247, 251, 273, 268, 281, 295, 308, 315, 317, 339, 340, 361, 353, 382, 381, 395, 407, 406, 427, 431, 452, 457, 469, 472

Offset

1,2

Comments

Sequence is not increasing: a(20)=135 and a(21)=131. Difference in the number of lattice points under the curve xy = n(n+1)/2 and xy = n(n-1)/2. - Emeric Deutsch, Aug 03 2005

Links

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Maple

with(numtheory): a:=n->add(tau(j), j=n*(n-1)/2+1..n*(n+1)/2): seq(a(n), n=1..64); # Emeric Deutsch, Aug 03 2005

Mathematica

Module[{nn=60, ds}, ds=DivisorSigma[0, Range[(nn(nn+1))/2]]; Table[Total[ Take[ ds, {(n(n-1))/2+1, (n(n+1))/2}]], {n, nn}]] (* Harvey P. Dale, Mar 14 2014 *)
With[{nn=60}, Total/@TakeList[DivisorSigma[0, Range[(nn(nn+1))/2]], Range[ nn]]] (* Harvey P. Dale, Mar 29 2022 *)

Prog

(PARI) a(n) = sum(k=n*(n-1)/2+1, n*(n+1)/2, numdiv(k)); \\ Michel Marcus, Aug 20 2019

Crossrefs

Cf. A000005 (tau).

Sequence in context: A312962 A312963 A312964 * A312965 A360403 A312966

Adjacent sequences: A090495 A090496 A090497 * A090499 A090500 A090501

Keyword

nonn

Author

Amarnath Murthy, Dec 04 2003

Extensions

Corrected and extended by Emeric Deutsch, Aug 03 2005

Status

approved