

A049822


a(n) = 1  tau(n) + Sum_{dn} tau(d1).


4



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
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OFFSET

1,5


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 nth 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.


LINKS

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


EXAMPLE

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}.


MATHEMATICA

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


PROG

(PARI) a(n) = 1  numdiv(n) + sumdiv(n, d, if (d==1, 0, numdiv(d1))); \\ Michel Marcus, Oct 01 2013


CROSSREFS

Column 3 of A122934.
Cf. A000005, A003238, A057427.
Cf. A069905 (number of partitions of n into 3 positive parts).
Sequence in context: A115980 A088936 A328405 * A140060 A164341 A333257
Adjacent sequences: A049819 A049820 A049821 * A049823 A049824 A049825


KEYWORD

easy,nonn


AUTHOR

Clark Kimberling


EXTENSIONS

Additional comments from Vladeta Jovovic, Aug 23 2003, Zak Seidov, Aug 31 2006 and Franklin T. AdamsWatters, Sep 20 2006
Edited by N. J. A. Sloane, Sep 21 2006


STATUS

approved



