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A066446
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Number of unordered divisor pairs of n.
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14
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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
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1,4
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LINKS
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FORMULA
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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).
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EXAMPLE
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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.
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MAPLE
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with(numtheory): seq(tau(n)*(tau(n)-1)/2, n=1..60); # Ridouane Oudra, Apr 15 2023
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MATHEMATICA
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Table[ Binomial[ DivisorSigma[0, n], 2], {n, 1, 100}]
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PROG
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(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'
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CROSSREFS
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KEYWORD
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easy,nonn
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AUTHOR
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STATUS
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approved
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