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 A049822 a(n) = 1 - tau(n) + Sum_{d|n} tau(d-1). 4

%I

%S 0,0,1,1,2,2,3,2,4,4,3,4,5,4,6,5,4,6,5,6,9,6,3,6,9,7,7,8,5,10,7,6,9,7,

%T 8,11,8,6,9,10,7,12,7,8,14,8,3,10,12,13,10,11,5,10,12,12,13,8,3,14,11,

%U 8,15,11,13,16,7,9,9,14,7,14,11,9,16,12,11,15,7,14,16,11,3,18,17,10,9,12

%N a(n) = 1 - tau(n) + Sum_{d|n} tau(d-1).

%C Number of partitions of n into 3 summands 0 < a <= b <= c with b/a and c/b integers.

%C a(n) is the number of 1's in the n-th row of array T given by A049816. E.g., there are 5 numbers k from 1 to 13 for which the Euclidean algorithm on (13, k) has exactly 1 nonzero remainder; hence a(13) = 5.

%H G. C. Greubel, <a href="/A049822/b049822.txt">Table of n, a(n) for n = 1..5000</a>

%e a(6) = 2 because of the 3 partitions of 6 into 3 parts, [4,1,1] and [2,2,2] meet the definition; [3,2,1] fails because 2 does not divide 3.

%e a(100) = 20 because there are 20 partitions of 100 in 3 summands 0 < a <= b <= c with integer b/a and c/b: {a, b, c} = {1, 1, 98}, {1, 3, 96}, {1, 9, 90}, {1, 11, 88}, {1, 33, 66}, {2, 2, 96}, {2, 14, 84}, {4, 4, 92}, {4, 8, 88}, {4, 12, 84}, {4, 16, 80}, {4, 24, 72}, {4, 32, 64}, {4, 48, 48}, {5, 5, 90}, {10, 10, 80}, {10, 30, 60}, {20, 20, 60}, {20, 40, 40}, {25, 25, 50}.

%t a[n_] := 1 - DivisorSigma[0, n] + DivisorSum[n, If[# == 1, 0, DivisorSigma[ 0, # - 1]]& ]; Array[a, 90] (* _Jean-François Alcover_, Dec 02 2015 *)

%o (PARI) a(n) = 1 - numdiv(n) + sumdiv(n, d, if (d==1, 0, numdiv(d-1))); \\ _Michel Marcus_, Oct 01 2013

%Y Column 3 of A122934.

%Y Cf. A000005, A003238, A057427.

%Y Cf. A069905 (number of partitions of n into 3 positive parts).

%K easy,nonn

%O 1,5

%A _Clark Kimberling_