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

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, 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, 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

Offset: 1

Author

Clark Kimberling

Keywords

• easy
• nonn

Comments

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

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.

Examples

			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.
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}.
		

Links

• G. C. Greubel, Table of n, a(n) for n = 1..5000

Crossrefs

Column 3 of A122934.

Cf. A000005, A003238, A057427.

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

Programs

Mathematica

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

PARI

a(n) = 1 - numdiv(n) + sumdiv(n, d, if (d==1, 0, numdiv(d-1))); \\ Michel Marcus, Oct 01 2013

Extensions

Additional comments from Vladeta Jovovic, Aug 23 2003, Zak Seidov, Aug 31 2006 and Franklin T. Adams-Watters, Sep 20 2006

Edited by N. J. A. Sloane, Sep 21 2006