A066446 - OEIS

A066446

Number of unordered divisor pairs of n.

0, 1, 1, 3, 1, 6, 1, 6, 3, 6, 1, 15, 1, 6, 6, 10, 1, 15, 1, 15, 6, 6, 1, 28, 3, 6, 6, 15, 1, 28, 1, 15, 6, 6, 6, 36, 1, 6, 6, 28, 1, 28, 1, 15, 15, 6, 1, 45, 3, 15, 6, 15, 1, 28, 6, 28, 6, 6, 1, 66, 1, 6, 15, 21, 6, 28, 1, 15, 6, 28, 1, 66, 1, 6, 15, 15, 6, 28, 1, 45, 10, 6, 1, 66, 6, 6, 6, 28

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OFFSET

1,4

LINKS

Antti Karttunen, Table of n, a(n) for n = 1..65537 (terms 1..1000 from Harry J. Smith)

FORMULA

a(p) = 1 iff p is a prime.

Combinations of d(n), the number of divisors of n (A000005), taken two at a time. If the canonical factorization of n into prime powers is Product p^e(p) then d(n) = Product (e(p) + 1). Therefore a(n) = C(d(n), 2) = d(n)*{ d(n)-1 }/2 which is a triangular number (A000217).

a(n) = A184389(n) - A000005(n) = A035116(n) - A184389(n). - Reinhard Zumkeller, Sep 08 2015

a(n) = A000217(A000005(n)-1). - Antti Karttunen, Sep 21 2018

a(n) = Sum_{k|n, i|n, i < k} 1. - Wesley Ivan Hurt, Aug 20 2020

a(n) = Sum_{d|n} A063647(d). - Ridouane Oudra, Apr 15 2023

EXAMPLE

The divisors of 6 are 1, 2, 3 & 6. In unordered pairs they are {1, 2}, {1, 3}, {1, 6}, {2, 3}, {2, 6}, & {3, 6}. Since there are six pairs, a(6) = 6. Also d(6) = 4. 4*3/2 = 6.

MAPLE

with(numtheory): seq(tau(n)*(tau(n)-1)/2, n=1..60); # Ridouane Oudra, Apr 15 2023

MATHEMATICA

Table[ Binomial[ DivisorSigma[0, n], 2], {n, 1, 100}]

PROG

(PARI) { for (n=1, 1000, a=binomial(numdiv(n), 2); write("b066446.txt", n, " ", a) ) } \\ Harry J. Smith, Feb 15 2010

(Haskell)

a066446 = a000217 . subtract 1 . a000005'

-- Reinhard Zumkeller, Sep 08 2015

CROSSREFS

Cf. A000005, A000217, A129510.

Cf. A035116, A184389, A063647.