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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). 1
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 (list; graph; refs; listen; history; text; internal format)
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

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Last modified August 23 13:45 EDT 2019. Contains 326227 sequences. (Running on oeis4.)