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A065608 Sum of divisors of n minus the number of divisors of n. 29
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.

In other words, number of ways to write n = (a + b) * k for positive integers a, b, k. - Gus Wiseman, Mar 25 2021

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

From Gus Wiseman, Mar 25 2021: (Start)

Also the number of compositions of n into an even number of parts with alternating parts equal. These are finite even-length sequences q of positive integers summing to n such that q(i) = q(i+2) for all possible i. For example, the a(2) = 1 through a(8) = 11 compositions are:

  (1,1)  (1,2)  (1,3)      (1,4)  (1,5)          (1,6)  (1,7)

         (2,1)  (2,2)      (2,3)  (2,4)          (2,5)  (2,6)

                (3,1)      (3,2)  (3,3)          (3,4)  (3,5)

                (1,1,1,1)  (4,1)  (4,2)          (4,3)  (4,4)

                                  (5,1)          (5,2)  (5,3)

                                  (1,2,1,2)      (6,1)  (6,2)

                                  (2,1,2,1)             (7,1)

                                  (1,1,1,1,1,1)         (1,3,1,3)

                                                        (2,2,2,2)

                                                        (3,1,3,1)

                                                        (1,1,1,1,1,1,1,1)

The odd-length version is A062968.

The version with alternating parts weakly decreasing is A114921, or A342528 if odd-length compositions are included.

The version with alternating parts unequal is A342532, or A224958 if odd-length compositions are included (unordered: A339404/A000726).

Allowing odd lengths as well as even gives A342527.

(End)

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).

Joerg Arndt, On computing the generalized Lambert series, arXiv:1202.6525v3 [math.CA], (2012).

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

G.f.: Sum_{n >= 1} q^(n^2)*( (n - 1) + q^n - (n - 1)*q^(2*n) )/(1 - q^n)^2 - differentiate equation 1 in Arndt with respect to t, then set x = q and t = q. - Peter Bala, Jan 22 2021

a(n) = A342527(n) - A062968(n). - Gus Wiseman, Mar 25 2021

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) a(n) = sigma(n) - numdiv(n); \\ 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.

A175342/A325545 count compositions with constant/distinct differences.

Cf. A001522, A002843, A008965, A064410, A064428, A325557, A342495.

Sequence in context: A337973 A223592 A180444 * A184396 A329718 A077764

Adjacent sequences:  A065605 A065606 A065607 * A065609 A065610 A065611

KEYWORD

nonn,easy

AUTHOR

Jason Earls, Nov 06 2001

STATUS

approved

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Last modified September 18 12:13 EDT 2021. Contains 347527 sequences. (Running on oeis4.)