A066446 -id:A066446 - cp's OEIS frontend

A343824 Sum of the elements in all pairs (d1, d2) of divisors of n such that d1<=d2, d1|n, d2|n, and d1 + d2 <= n.

0, 2, 2, 9, 2, 24, 2, 28, 12, 32, 2, 96, 2, 40, 36, 75, 2, 126, 2, 132, 44, 56, 2, 288, 18, 64, 52, 168, 2, 336, 2, 186, 60, 80, 52, 495, 2, 88, 68, 400, 2, 432, 2, 240, 198, 104, 2, 760, 24, 258, 84, 276, 2, 528, 68, 512, 92, 128, 2, 1296, 2, 136, 246, 441, 76, 624, 2, 348

Offset: 1

Wesley Ivan Hurt, Apr 30 2021

nonn

If n is prime, then a(n) = 2.

			a(7) = 2; There is one divisor pair of 7 whose sum is less than or equal to 7: (1,1). The sum is then 1+1 = 2.
a(9) = 12; The divisor pairs of 9 whose sum is less than or equal to 9 are: (1,1), (1,3) and (3,3). The sum of the coordinates is then (1+1) + (1+3) + (3+3) = 12.

Antti Karttunen, Table of n, a(n) for n = 1..20000

Cf. A066446.

Table[Sum[Sum[(i + k) (1 - Ceiling[n/k] + Floor[n/k]) (1 - Ceiling[n/i] + Floor[n/i]), {i, k}], {k, Floor[n/2]}], {n, 80}]

a(n) = sumdiv(n, d1, sumdiv(n, d2, if ((d1 <= d2) && (d1+d2 <= n), d1+d2))); \\ Michel Marcus, May 01 2021

a(n) = Sum_{k=1..floor(n/2)} Sum_{i=1..k} c(n/k) * c(n/i) * (i+k), where c(n) = 1 - ceiling(n) + floor(n).

A337524 a(n) = d(n) * (d(n) - 1), where d is the number of divisors of n (A000005).

Original entry on oeis.org

0, 2, 2, 6, 2, 12, 2, 12, 6, 12, 2, 30, 2, 12, 12, 20, 2, 30, 2, 30, 12, 12, 2, 56, 6, 12, 12, 30, 2, 56, 2, 30, 12, 12, 12, 72, 2, 12, 12, 56, 2, 56, 2, 30, 30, 12, 2, 90, 6, 30, 12, 30, 2, 56, 12, 56, 12, 12, 2, 132, 2, 12, 30, 42, 12, 56, 2, 30, 12, 56, 2, 132, 2, 12, 30, 30

Offset: 1

Wesley Ivan Hurt, Aug 30 2020

Number of distinct nonsquare m X n matrices such that m and n are divisors of n.

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

Cf. A000005 (tau).

Equals twice A066446.

Table[DivisorSigma[0, n]^2 - DivisorSigma[0, n], {n, 100}]
#(#-1)&/@DivisorSigma[0,Range[100]] (* Harvey P. Dale, Nov 03 2022 *)

A373113 For n >= 1, a(n) = Sum_{i = n..(n + A000005(n) - 1)} i.

Original entry on oeis.org

1, 5, 7, 15, 11, 30, 15, 38, 30, 46, 23, 87, 27, 62, 66, 90, 35, 123, 39, 135, 90, 94, 47, 220, 78, 110, 114, 183, 59, 268, 63, 207, 138, 142, 146, 360, 75, 158, 162, 348, 83, 364, 87, 279, 285, 190, 95, 525, 150, 315, 210, 327, 107, 460, 226, 476, 234, 238, 119, 786, 123

Offset: 1

Ctibor O. Zizka, May 25 2024

nonn

A005385 is a subsequence.

			n = 4: A000005(4) = 3, 4 + 5 + 6 = 15, thus a(4) = 15.
n = 5: A000005(5) = 2, 5 + 6 = 11, thus a(5) = 11.
n = 6: A000005(6) = 4, 6 + 7 + 8 + 9 = 30, thus a(6) = 30.

Cf. A000005, A005384, A005385, A066446.

a[n_] := Module[{d = DivisorSigma[0, n]}, d*(d-1)/2 + n*d]; Array[a, 60] (* Amiram Eldar, May 25 2024 *)

a(n) = my(d=numdiv(n)); n*d + d*(d-1)/2; \\ Michel Marcus, May 28 2024

a(n) = A066446(n) + n*A000005(n).

a(n) = A000005(n)*(A000005(n) - 1)/2 + n*A000005(n).

cp's OEIS Frontend

A343824 Sum of the elements in all pairs (d1, d2) of divisors of n such that d1<=d2, d1|n, d2|n, and d1 + d2 <= n.

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A337524 a(n) = d(n) * (d(n) - 1), where d is the number of divisors of n (A000005).

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A373113 For n >= 1, a(n) = Sum_{i = n..(n + A000005(n) - 1)} i.

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