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A065608 Sum of divisors of n minus the number of divisors of n. 19
0, 1, 2, 4, 4, 8, 6, 11, 10, 14, 10, 22, 12, 20, 20, 26, 16, 33, 18, 36, 28, 32, 22, 52, 28, 38, 36, 50, 28, 64, 30, 57, 44, 50, 44, 82, 36, 56, 52, 82, 40, 88, 42, 78, 72, 68, 46, 114, 54, 87, 68, 92, 52, 112, 68, 112, 76, 86, 58, 156, 60, 92, 98, 120, 80, 136, 66, 120, 92 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

Number of permutations p of {1,2,...,n} such that p(k)-k takes exactly two distinct values. Example: a(4)=4 because we have 4123, 3412, 2143 and 2341. - Max Alekseyev and Emeric Deutsch, Dec 22 2006

Number of solutions to the Diophantine equation xy + yz = n, with x,y,z >= 1.

Not the same as A184396(n): a(66) = 136 while A184396(66) = 137. - Wesley Ivan Hurt, Dec 26 2013

LINKS

Alois P. Heinz, Table of n, a(n) for n = 1..20000 (first 1000 terms from T. D. Noe)

M. Alekseyev, E. Deutsch, and J. H. Steelman, Solution to problem 11281, Amer. Math. Monthly, 116, No. 5, 2009, p. 465.

George E. Andrews, Stacked lattice boxes, Ann. Comb. 3 (1999), 115-130. See L_2(n).

FORMULA

a(n) = sigma(n) - d(n) = A000203(n) - A000005(n).

a(n) = Sum_{d|n} (d-1). - Wesley Ivan Hurt, Dec 26 2013

G.f.: Sum_{k>=1} x^(2*k)/(1-x^k)^2. - Benoit Cloitre, Apr 21 2003

G.f.: Sum_{n>=1} (n-1)*x^n/(1-x^n). - Joerg Arndt, Jan 30 2011

L.g.f.: -log(Product_{k>=1} (1 - x^k)^(1-1/k)) = Sum_{n>=1} a(n)*x^n/n. - Ilya Gutkovskiy, Mar 18 2018

MAPLE

with(numtheory): seq(sigma(n)-tau(n), n=1..70); # Emeric Deutsch, Dec 22 2006

MATHEMATICA

Table[DivisorSigma[1, n]-DivisorSigma[0, n], {n, 100}] (* Wesley Ivan Hurt, Dec 26 2013 *)

PROG

(PARI) { for (n = 1, 1000, a=sigma(n) - numdiv(n); write("b065608.txt", n, " ", a) ) } \\ Harry J. Smith, Oct 23 2009

(GAP) List([1..100], n->Sigma(n)-Tau(n)); # Muniru A Asiru, Mar 19 2018

CROSSREFS

Cf. A000203, A000005, A134857.

Starting (1, 2, 4, 4, 8, 6, ...), = row sums of triangle A077478. - Gary W. Adamson, Nov 12 2007

Starting with "1" = row sums of triangle A176919. - Gary W. Adamson, Apr 29 2010

Column k=2 of A125182.

Sequence in context: A079890 A223592 A180444 * A184396 A077764 A110794

Adjacent sequences:  A065605 A065606 A065607 * A065609 A065610 A065611

KEYWORD

nonn,changed

AUTHOR

Jason Earls, Nov 06 2001

STATUS

approved

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Last modified February 20 15:20 EST 2019. Contains 320336 sequences. (Running on oeis4.)