A299251 a(n) = ((Sum_{k=1..floor((n+1)^2/4)} d(k)) - T(n)) / 2, where d(n) = number of divisors of n (A000005) and T(n) = the n-th triangular number (A000217).

0, 0, 1, 2, 4, 7, 11, 15, 21, 28, 37, 45, 55, 67, 80, 95, 110, 127, 146, 164, 187, 209, 235, 260, 286, 315, 346, 380, 413, 449, 485, 522, 564, 605, 651, 695, 743, 792, 844, 898, 950, 1006, 1064, 1123, 1185, 1250, 1318, 1384, 1451, 1523, 1596, 1670, 1747, 1828

Offset

1,4

Comments

Twice this sequence is an attempt to find a counterpart to A161664: both compare triangular numbers T(n) and partial sums of numbers of divisors S(n). A161664 computes the excess of T(n) compared to S(n), whereas 2*a(n) computes the excess of S(n') compared to T(n), where n' is chosen equal to floor((n+1)^2/4). This choice appears structurally natural and economical when illustrated in a diagram. (See provided link.)

Links

Table of n, a(n) for n=1..54.

Luc Rousseau, Accompanying diagram

Formula

a(n) = (A006218(A002620(n + 1)) - A000217(n)) / 2.

Mathematica

F[n_] := Floor[(1/4)*n^2]
A[n_] := (Sum[DivisorSigma[0, k], {k, 1, F[n + 1]}] - n*(n + 1)/2)/2
Table[A[n], {n, 1, 100}]

Prog

(PARI)
f(n)=floor(n^2/4)
a(n)=(sum(k=1, f(n+1), numdiv(k))-n*(n+1)/2)/2
for(n=1, 100, print1(a(n), ", "))

Crossrefs

Cf. A000005, A000217, A002620, A006218, A161664.

Sequence in context: A293239 A261878 A261993 * A238485 A316264 A067744

Adjacent sequences: A299248 A299249 A299250 * A299252 A299253 A299254

Keyword

nonn

Author

Luc Rousseau, Feb 06 2018

Status

approved