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A003165
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a(n) = floor(n/2) + 1 - d(n), where d(n) is the number of divisors of n.
(Formerly M0106)
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2
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0, 0, 0, 0, 1, 0, 2, 1, 2, 2, 4, 1, 5, 4, 4, 4, 7, 4, 8, 5, 7, 8, 10, 5, 10, 10, 10, 9, 13, 8, 14, 11, 13, 14, 14, 10, 17, 16, 16, 13, 19, 14, 20, 17, 17, 20, 22, 15, 22, 20, 22, 21, 25, 20, 24, 21, 25, 26, 28, 19, 29, 28, 26, 26, 29, 26, 32, 29, 31, 28, 34, 25, 35, 34, 32
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OFFSET
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1,7
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COMMENTS
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a(n) is the number of partitions of n into exactly two parts whose smallest part is a nondivisor of n (see example). If n is prime, all of the smallest parts (except for 1) are nondivisors of n. Since there are floor(n/2) total partitions of n into two parts, then a(n) = floor(n/2) - 1 for primes (since we exclude 1). Proof: n = p implies a(p) = floor(p/2) + 1 - d(p) = floor(p/2) + 1 - 2 = floor(p/2) - 1. Furthermore, if n is an odd prime, a(n) = (n-3)/2. - Wesley Ivan Hurt, Jul 16 2014
a(n) is the nullity of the (n-1) X (n-1) matrix M(n) with entries M(n)[i,j] = i*j mod n (matrices given by A352620). - Luca Onnis, Mar 27 2022
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REFERENCES
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M. Newman, Integral Matrices. Academic Press, NY, 1972, p. 186.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
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FORMULA
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a(n) = Sum_{i=1..floor(n/2)} ceiling(n/i) - floor(n/i). - Wesley Ivan Hurt, Jul 16 2014
a(n) = Sum_{i=1..n} ceiling(n/i) mod floor(n/i). - Wesley Ivan Hurt, Sep 15 2017
G.f.: x*(1 + x - x^2)/((1 - x)^2*(1 + x)) - Sum_{k>=1} x^k/(1 - x^k). - Ilya Gutkovskiy, Sep 18 2017
a(n) = Sum_{i=1..floor((n-1)/2)} sign((n-i) mod i). - Wesley Ivan Hurt, Nov 17 2017
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EXAMPLE
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a(20) = 5. The partitions of 20 into exactly two parts are: (19,1), (18,2), (17,3), (16,4), (15,5), (14,6), (13,7), (12,8), (11,9), (10,10). Of these, there are exactly 5 partitions whose smallest part does not divide 20: {3,6,7,8,9}. - Wesley Ivan Hurt, Jul 16 2014
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MAPLE
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MATHEMATICA
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Table[Floor[n/2]+1-DivisorSigma[0, n], {n, 80}] (* Harvey P. Dale, May 09 2011 *)
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PROG
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(Sage)
return sum(1 for k in (1..n//2) if n % k)
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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