

A001065


Sum of proper divisors (or aliquot parts) of n: sum of divisors of n that are less than n.
(Formerly M2226 N0884)


493



0, 1, 1, 3, 1, 6, 1, 7, 4, 8, 1, 16, 1, 10, 9, 15, 1, 21, 1, 22, 11, 14, 1, 36, 6, 16, 13, 28, 1, 42, 1, 31, 15, 20, 13, 55, 1, 22, 17, 50, 1, 54, 1, 40, 33, 26, 1, 76, 8, 43, 21, 46, 1, 66, 17, 64, 23, 32, 1, 108, 1, 34, 41, 63, 19, 78, 1, 58, 27, 74, 1, 123, 1, 40, 49, 64, 19, 90, 1, 106
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OFFSET

1,4


COMMENTS

Also total number of parts in all partitions of n into equal parts that do not contain 1 as a part.  Omar E. Pol, Jan 16 2013
Related concepts: If a(n) < n, n is said to be deficient, if a(n) > n, n is abundant, and if a(n) = n, n is perfect. If there is a cycle of length 2, so that a(n) = b and a(b) = n, b and n are said to be amicable. If there is a longer cycle, the numbers in the cycle are said to be sociable. See examples.  Juhani Heino, Jul 17 2017
Sum of the smallest parts in the partitions of n into two parts such that the smallest part divides the largest.  Wesley Ivan Hurt, Dec 22 2017
a(n) is also the total number of parts congruent to 0 mod k in the partitions of k*n into equal parts that do not contain k as a part (the comment dated Jan 16 2013 is the case for k = 1).  Omar E. Pol, Nov 23 2019


REFERENCES

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 840.
George E. Andrews, Number Theory. New York: Dover, 1994; Pages 1, 7592; p. 92 #15: Sigma(n) / d(n) >= n^(1/2).
Carl Pomerance, The first function and its iterates, pp. 125138 in Connections in Discrete Mathematics, ed. S. Butler et al., Cambridge, 2018.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).


LINKS

T. D. Noe, Table of n, a(n) for n = 1..10000
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972. [alternative scanned copy].
Joerg Arndt, On computing the generalized Lambert series, arXiv:1202.6525v3 [math.CA], (2012).
Henry Bottomley, Illustration of initial terms
Don Coppersmith, An answer to the problem of Don Saari, 1987.
Paul Erdős, Andrew Granville, Carl Pomerance and Claudia Spiro, On the normal behavior of the iterates of some arithmetic functions, Analytic number theory, Birkhäuser Boston, 1990, pp. 165204.
Paul Erdős, Andrew Granville, Carl Pomerance and Claudia Spiro, On the normal behavior of the iterates of some arithmetic functions, Analytic number theory, Birkhäuser Boston, 1990, pp. 165204. [Annotated copy with Anumbers]
P. Pollack and C. Pomerance, Some problems of Erdős on the sumofdivisors function, For Richard Guy on his 99th birthday: May his sequence be unbounded, Trans. Amer. Math. Soc. Ser. B 3 (2016), 126; errata.
Carl Pomerance and HeeSung Yang, Variant of a theorem of Erdős on the sumofproperdivisors function, Math. Comp., to appear (2014).
Primefan, Sums of Restricted Divisors for n=1 to 1000
F. Richman, Aliquot series: Abundant, deficient, perfect
Eric Weisstein's World of Mathematics, Restricted Divisor Function
Eric Weisstein's World of Mathematics, Divisor Function
Index entries for "core" sequences


FORMULA

G.f.: Sum_{k>0} k * x^(2*k)/(1  x^k).  Michael Somos, Jul 05 2006
a(n) = sigma(n)  n = A000203(n)  n.  Lekraj Beedassy, Jun 02 2005
a(n) = A155085(n).  Michael Somos, Sep 20 2011
Equals inverse Mobius transform of A051953 = A051731 * A051953. Example: a(6) = 6 = (1, 1, 1, 0, 0, 1) dot (0, 1, 1, 2, 1, 4) = (0 + 1 + 1 + 0 + 0 + 4), where A051953 = (0, 1, 1, 2, 1, 4, 1, 4, 3, 6, 1, 8, ...) and (1, 1, 1, 0, 0, 1) = row 6 of A051731 where the 1's positions indicate the factors of 6.  Gary W. Adamson, Jul 11 2008
a(n) = A006128(n)  A220477(n)  n.  Omar E. Pol Jan 17 2013
a(n) = Sum_{i=1..floor(n/2)} i*(1ceiling(frac(n/i))).  Wesley Ivan Hurt, Oct 25 2013
a(n) = n  A033879(n) = n + A033880(n).  Omar E. Pol, Dec 30 2013
Dirichlet g.f.: zeta(s1)*(zeta(s)  1).  Ilya Gutkovskiy, Aug 07 2016
a(n) = 1 + A048050(n), n > 1.  R. J. Mathar, Mar 13 2018
Erdős (Elem. Math. 28 (1973), 8386) shows that the density of even integers in the range of a(n) is strictly less than 1/2. The argument of Coppersmith (1987) shows that the range of a(n) has density at most 47/48 < 1.  N. J. A. Sloane, Dec 21 2019
G.f.: Sum_{k >= 2} x^k/(1  x^k)^2. Cf. A296955. (This follows from the fact that if g(z) = Sum_{n >= 1} a(n)*z^n and f(z) = Sum_{n >= 1} a(n)*z^(N*n)/(1  z^n) then f(z) = Sum_{k >= N} g(z^k), taking a(n) = n and N = 2.)  Peter Bala, Jan 13 2021
Faster converging g.f.: Sum_{n >= 1} q^(n*(n+1))*(n*q^(3*n+2)  (n + 1)*q^(2*n+1)  (n  1)*q^(n+1) + n)/((1  q^n)*(1  q^(n+1))^2). (In equation 1 in Arndt, after combining the two n = 0 summands to get t/(1  t), apply the operator t*d/dt to the resulting equation and then set t = q and x = 1.)  Peter Bala, Jan 22 2021
a(n) = Sum_{dn} d * (1  [n = d]), where [ ] is the Iverson bracket.  Wesley Ivan Hurt, Jan 28 2021
a(n) = Sum_{i=1..n} ((n1) mod i)  (n mod i). [See also A176079.]  José de Jesús Camacho Medina, Feb 23 2021


EXAMPLE

x^2 + x^3 + 3*x^4 + x^5 + 6*x^6 + x^7 + 7*x^8 + 4*x^9 + 8*x^10 + x^11 + ...
For n = 44, sum of divisors of n = sigma(n) = 84; so a(44) = 8444 = 40.
Related concepts: (Start)
From 1 to 17, all n are deficient, except 6 and 12 seen below. See A005100.
Abundant numbers: a(12) = 16, a(18) = 21. See A005101.
Perfect numbers: a(6) = 6, a(28) = 28. See A000396.
Amicable numbers: a(220) = 284, a(284) = 220. See A259180.
Sociable numbers: 12496 > 14288 > 15472 > 14536 > 14264 > 12496. See A122726. (End)
For n = 10 the sum of the divisors of 10 that are less than 10 is 1 + 2 + 5 = 8. On the other hand, the partitions of 10 into equal parts that do not contain 1 as a part are [10], [5,5], [2,2,2,2,2], there are 8 parts, so a(10) = 8.  Omar E. Pol, Nov 24 2019


MAPLE

A001065 := proc(n)
numtheory[sigma](n)n ;
end proc:
seq( A001065(n), n=1..100) ;


MATHEMATICA

Table[ Plus @@ Select[ Divisors[ n ], #<n & ], {n, 1, 90} ]
Table[Plus @@ Divisors[n]  n, {n, 1, 90}] (* Zak Seidov, Sep 10 2009 *)
Table[DivisorSigma[1, n]  n, {n, 1, 80}] (* JeanFrançois Alcover, Apr 25 2013 *)
Array[Plus @@ Most@ Divisors@# &, 80] (* Robert G. Wilson v, Dec 24 2017 *)


PROG

(PARI) {a(n) = if( n==0, 0, sigma(n)  n)} /* Michael Somos, Sep 20 2011 */
(MuPAD) numlib::sigma(n)n$ n=1..81 // Zerinvary Lajos, May 13 2008
(Haskell)
a001065 n = a000203 n  n  Reinhard Zumkeller, Sep 15 2011
(Magma) [SumOfDivisors(n)n: n in [1..100]]; // Vincenzo Librandi, May 06 2015
(Python)
from sympy import divisor_sigma
def A001065(n): return divisor_sigma(n)n # Chai Wah Wu, Nov 04 2022


CROSSREFS

Row sums of A141846.  Gary W. Adamson, Jul 11 2008
Row sums of A176891.  Gary W. Adamson, May 02 2010
Row sums of A176079.  Mats Granvik, May 20 2012
a(n) = sum (A027751(n,k): k = 1..A000005(n)1).  Reinhard Zumkeller, Apr 05 2013
Alternating row sums of A231347.  Omar E. Pol, Jan 02 2014
For n > 1: a(n) = A240698(n,A000005(n)1).  Reinhard Zumkeller, Apr 10 2014
A134675(n) = A007434(n) + a(n).  Conjectured by John Mason and proved by Max Alekseyev, Jan 07 2015
Cf. A032741, A000203, A048050, A000593, A034090, A034091, A027750.
Cf. A051953, A051731.
Cf. A037020 (primes), A053868, A053869 (odd and even terms).
Cf. A048138 (number of occurrences), A238895, A238896 (record values thereof).
Cf. A007956 (products of proper divisors).
Cf. A005100, A005101, A000396, A259180, A122726 (related concepts).
Sequence in context: A357698 A069250 A294888 * A173455 A324535 A318501
Adjacent sequences: A001062 A001063 A001064 * A001066 A001067 A001068


KEYWORD

nonn,core,easy,nice


AUTHOR

N. J. A. Sloane, R. K. Guy


STATUS

approved



