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A000203 sigma(n) = sum of divisors of n. Also called sigma_1(n).
(Formerly M2329 N0921)
1947
1, 3, 4, 7, 6, 12, 8, 15, 13, 18, 12, 28, 14, 24, 24, 31, 18, 39, 20, 42, 32, 36, 24, 60, 31, 42, 40, 56, 30, 72, 32, 63, 48, 54, 48, 91, 38, 60, 56, 90, 42, 96, 44, 84, 78, 72, 48, 124, 57, 93, 72, 98, 54, 120, 72, 120, 80, 90, 60, 168, 62, 96, 104, 127, 84, 144, 68, 126, 96, 144 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

Multiplicative: If the canonical factorization of n into prime powers is the product of p^e(p) then sigma_k(n) = Product_p ((p^((e(p)+1)*k))-1)/(p^k-1).

Sum_{d|n} 1/d^k is equal to sigma_k(n)/n^k. So sequences A017665-A017712 also give the numerators and denominators of sigma_k(n)/n^k for k = 1..24. The power sums sigma_k(n) are in sequences A000203 (this sequence) (k=1), A001157-A001160 (k=2,3,4,5), A013954-A013972 for k = 6,7,...,24. - Ahmed Fares (ahmedfares(AT)my-deja.com), Apr 05 2001

A number n is abundant if sigma(n) > 2n (cf. A005101), perfect if sigma(n) = 2n (cf. A000396), deficient if sigma(n) < 2n (cf. A005100).

a(n) = number of sublattices of index n in a generic 2-dimensional lattice. - Avi Peretz (njk(AT)netvision.net.il), Jan 29 2001

The sublattices of index n are in one-to-one correspondence with matrices [a b; 0 d] with a>0, ad=n, b in [0..d-1]. The number of these is Sum_{d|n} = sigma(n), which is A000203. A sublattice is primitive if gcd(a,b,d) = 1; the number of these is n * product_{p|n} (1+1/p), which is A001615. [Cf. Grady reference.]

Sum of number of common divisors of n and m, where m runs from 1 to n. - Naohiro Nomoto, Jan 10 2004

a(n) is the cardinality of all extensions over Q_p with degree n in the algebraic closure of Q_p, where p>n. - Volker Schmitt (clamsi(AT)gmx.net), Nov 24 2004. Cf. A100976, A100977, A100978 (p-adic extensions).

s(n) = s(n-1) + s(n-2) - s(n-5) - s(n-7) + s(n-12) + s(n-15) - s(n-22) - s(n-26) + ... if n is not pentagonal, i.e. n != (3 j^2 +- j)/2, and the sum is instead s(n) + ((-1)^j))*n if n is pentagonal. - Gary W. Adamson, Oct 05 2008 [corrected Apr 27 2012 by William J. Keith based on Ewell]

Prefaced with a zero: (0, 1, 3, 4, 7,...) = A147843 convolved with the partition numbers, A000041. - Gary W. Adamson, Nov 15 2008

Write n as 2^k * d, where d is odd. Then a(n) is odd if and only if d is a square. - Jon Perry, Nov 08 2012

Also total number of parts in the partitions of n into equal parts. - Omar E. Pol, Jan 16 2013

Note that sigma(3^4)=11^2. On the other hand, Kanold (1947) shows that the equation sigma(q^{p-1}) = b^p has no solutions b >2, q prime, p odd prime. - N. J. A. Sloane, Dec 21 2013, based on postings to the Number Theory Mailing List by Vladimir Letsko and Luis H. Gallardo

a(n) = A240698(n, A000005(n)). - Reinhard Zumkeller, Apr 10 2014

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.

T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, page 38.

A. T. Benjamin and J. J. Quinn, Proofs that really count: the art of combinatorial proof, M.A.A. 2003, p. 116ff.

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 162, #16, (6), 2nd formula.

J. W. L. Glaisher, On the function chi(n), Quarterly Journal of Pure and Applied Mathematics, 20 (1884), 97-167.

Ross Honsberger, "Mathematical Gems, Number One," The Dolciani Mathematical Expositions, Published and Distributed by The Mathematical Association of America, page 116.

Kanold, Hans Joachim, Kreisteilungspolynome und ungerade vollkommene Zahlen. (German), Ber. Math.-Tagung Tübingen 1946, (1947). pp. 84-87.

M. Krasner, Le nombre des surcorps primitifs d'un degre donne et le nombre des surcorps metagaloisiens d'un degre donne d'un corps de nombres p-adiques. Comptes Rendus Hebdomadaires, Academie des Sciences, Paris 254, 255, 1962

A. Lubotzky, Counting subgroups of finite index, Proceedings of the St. Andrews/Galway 93 group theory meeting, Th. 2.1. LMS Lecture Notes Series no. 212 Cambridge University Press 1995.

G. Polya, Induction and Analogy in Mathematics, vol. 1 of Mathematics and Plausible Reasoning, Princeton Univ Press 1954, page 92.

Carl Pomerance and Hee-Sung Yang, Variant of a theorem of Erdos on the sum-of-proper-divisors function, Math. Comp., to appear (2014); http://www.math.dartmouth.edu/~carlp/uupaper7.pdf

J. S. Rutherford, The enumeration and symmetry-significant properties of derivative lattices, Acta Cryst. A48 (1992), 500-508. [From N. J. A. Sloane, Mar 14 2009]

J. S. Rutherford, The enumeration and symmetry-significant properties of derivative lattices II, Acta Cryst. A49 (1993), 293-300. [From N. J. A. Sloane, Mar 14 2009]

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).

Robert M. Young, Excursions in Calculus, The Mathematical Association of America, 1992 p.361 [From Gary W. Adamson, Oct 05 2008]

LINKS

N. J. A. Sloane and Daniel Forgues, Table of n, a(n) for n = 1..100000 (first 20000 terms from N. J. A. Sloane)

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].

B. Apostol, L. Petrescu, Extremal Orders of Certain Functions Associated with Regular Integers (mod n), Journal of Integer Sequences, 2013, # 13.7.5.

M. Baake and U. Grimm, Quasicrystalline combinatorics

H. Bottomley, Illustration of initial terms

C. K. Caldwell, The Prime Glossary, sigma function

J. N. Cooper and A. W. N. Riasanovsky, On the Reciprocal of the Binary Generating Function for the Sum of Divisors, 2012. - From N. J. A. Sloane, Dec 25 2012

L. Euler, Observatio de summis divisorum

L. Euler, An observation on the sums of divisors

J. A. Ewell, Recurrences for the sum of divisors, Proc. Amer. Math. Soc. 64 (2) 1977.

Daniele A. Gewurz and Francesca Merola, Sequences realized as Parker vectors ..., J. Integer Seqs., Vol. 6, 2003.

M. J. Grady, A group theoretic approach to a famous partition formula, Amer. Math. Monthly, 112 (No. 7, 2005), 645-651.

P. A. MacMahon, Divisors of numbers and their continuations in the theory of partitions, Proc. London Math. Soc., 19 (1921), 75-113.

M. Maia and M. Mendez, On the arithmetic product of combinatorial species, arXiv:math.CO/0503436

K. Matthews, Factorizing n and calculating phi(n),omega(n),d(n),sigma(n) and mu(n)

Walter Nissen, Abundancy : Some Resources

Jon Perry, More Partition Functions

John S. Rutherford, Sublattice enumeration. IV. Equivalence classes of plane sublattices by parent Patterson symmetry and colour lattice group type, Acta Cryst. (2009). A65, 156-163. [See Table 1]. - From N. J. A. Sloane, Feb 23 2009

Eric Weisstein's World of Mathematics, Divisor Function

Wikipedia, Divisor function

Index entries for sequences related to sublattices

Index entries for "core" sequences

FORMULA

Multiplicative with a(p^e) = (p^(e+1)-1)/(p-1). - David W. Wilson, Aug 01 2001

G.f.: -x*deriv(eta(x))/eta(x) where eta(x) = prod(n>=1, 1-x^n). - Joerg Arndt, Mar 14 2010

L.g.f.: -log(prod(j>=1 ,1-x^j)) = sum(n>=1, a(n)/n*x^n). - Joerg Arndt, Feb 04 2011

Dirichlet convolution of phi(n) and tau(n), i.e., a(n) = sum_{d|n} phi(n/d)*tau(d), cf. A000010, A000005.

a(n) is odd iff n is a square or twice a square. - Robert G. Wilson v, Oct 03 2001

a(n) = a(n*p(n)) - p(n)*a(n). - Labos Elemer, Aug 14 2003

a(n) = n*A000041(n) - sum{i=1, n-1, a(i)*A000041(n-i)}. - Jon Perry, Sep 11 2003

a(n) = -A010815(n)*n - Sum(A010815(k)*a(n-k): 1<=k<n). - Reinhard Zumkeller, Nov 30 2003

a(n) = f(n, 1, 1, 1), where f(n, i, x, s) = if n = 1 then s*x else if p(i)|n then f(n/p(i), i, 1+p(i)*x, s) else f(n, i+1, 1, s*x) with p(i) = i-th prime (A000040). - Reinhard Zumkeller, Nov 17 2004

Recurrence: n^2*(n-1)*a(n) = 12*Sum[(5*k*(n-k)-n^2)*a(k)*a(n-k), k=1..(n-1)] if n>1. - Dominique Giard (dominique.giard(AT)gmail.com), Jan 11 2005

G.f.: Sum_{k>0} k * x^k / (1 - x^k) = Sum_{k>0} x^k /(1 - x^k)^2. Dirichlet g.f.: zeta(s)*zeta(s-1). - Michael Somos, Apr 05 2003

For odd n, a(n) = A000593(n) sum of odd divisors of n. For even n, a(n) = A000593(n) + A074400(n/2) where A074400 is sum of the even divisors of 2n. - Jonathan Vos Post, Mar 26 2006

Equals the inverse Moebius transform of the natural numbers. Equals row sums of A127093. - Gary W. Adamson, May 20 2007

A127093 * [1/1, 1/2, 1/3,...] = [1/1, 3/2, 4/3, 7/4, 6/5, 12/6, 8/7,...]. Row sums of triangle A135539. - Gary W. Adamson, Oct 31 2007

Row sums of triangle A134838. - Gary W. Adamson, Nov 12 2007

a(n) = A054785(2*n) - A000593(2*n). - Reinhard Zumkeller, Apr 23 2008

a(n) = n*sum{k=1,n} A060642(n,k)/k*(-1)^(k+1). - Vladimir Kruchinin, Aug 10 2010

Dirichlet convolution of A037213 and A034448. - R. J. Mathar, Apr 13 2011

G.f.: A(x) = x/(1-x)*(1 - 2*x*(1-x)/(G(0) - 2*x^2 + 2*x)); G(k) = - 2*x - 1 - (1+x)*k + (2*k+3)*(x^(k+2)) - x*(k+1)*(k+3)*((-1 + (x^(k+2)))^2)/G(k+1); (continued fraction). - Sergei N. Gladkovskii, Dec 06 2011

a(n) = A001065(n) + n. - Mats Granvik, May 20 2012

a(n) = A006128(n) - A220477(n). - Omar E. Pol, Jan 17 2013

a(n) = sum_{k=1..A003056(n)} (-1)^(k-1)*A196020(n,k). - conjectured by Omar E. Pol and proved by Max Alekseyev, Nov 19 2013

a(n) = sum_{k=1..A003056(n)} (-1)^(k-1)*A000330(k)*A000716(n-A000217(k)). - Mircea Merca, Mar 05 2014

It appears that a(n) < 6*n^(3/2)/Pi^2 for n > 12. - Robert G. Wilson v, May 14 2014

EXAMPLE

For example, 6 is divisible by 1, 2, 3 and 6, so sigma(6) = 1 + 2 + 3 + 6 = 12.

Let L = <V,W> be a 2-dimensional lattice. The 7 sublattices of index 4 are generated by <4V,W>, <V,4W>, <4V,W+-V>, <2V,2W>, <2V+W,2W>, <2V,2W+V>. Compare A001615.

MAPLE

with(numtheory): A000203 := n->sigma(n); seq(A000203(n), n=1..100);

MATHEMATICA

Table[ DivisorSigma[1, n], {n, 1, 100}]

a[ n_] := SeriesCoefficient[ QPolyGamma[ 1, 1, q] / Log[q]^2, {q, 0, n}]; (* Michael Somos, Apr 25 2013 *)

PROG

(MAGMA) [ SumOfDivisors(n) : n in [1..40]];

(PARI) {a(n) = if( n<1, 0, sigma(n))};

(PARI) {a(n) = if( n<1, 0, direuler( p=2, n, 1 / (1 - X) /(1 - p*X))[n])};

(PARI) {a(n) = if( n<1, 0, polcoeff( sum( k=1, n, x^k / (1 - x^k)^2, x * O(x^n)), n))}; /* Michael Somos, Jan 29 2005 */

(MuPad) numlib::sigma(n)$ n=1..81 // Zerinvary Lajos, May 13 2008

(Sage) [sigma(n, 1) for n in xrange(1, 71)] # Zerinvary Lajos, Jun 04 2009

(PARI) max_n = 30; ser = - sum(k=1, max_n, log(1-x^k)) a(n) = polcoeff(ser, n)*n \\ Gottfried Helms, Aug 10 2009

(Maxima) makelist(divsum(n), n, 1, 1000); \\ Emanuele Munarini, Mar 26 2011

(Haskell)

a000203 n = product $ zipWith (\p e -> (p^(e+1)-1) `div` (p-1)) (a027748_row n) (a124010_row n)

-- Reinhard Zumkeller, May 07 2012

(MIT/GNU Scheme, with Aubrey Jaffer's SLIB Scheme library)

(require 'factor)

(define (A000203 n) (fold-left (lambda (prod p.e) (* prod (/ (- (expt (car p.e) (+ 1 (cdr p.e))) 1) (- (car p.e) 1)))) 1 (if (= 1 n) (list) (elemcountpairs (sort (factor n) <)))))

(define (elemcountpairs lista) (let loop ((pairs (list)) (lista lista) (prev #f)) (cond ((not (pair? lista)) (reverse! pairs)) ((equal? (car lista) prev) (set-cdr! (car pairs) (+ 1 (cdar pairs))) (loop pairs (cdr lista) prev)) (else (loop (cons (cons (car lista) 1) pairs) (cdr lista) (car lista))))))

;; Antti Karttunen, Dec 02 2013

CROSSREFS

See A034885, A002093 for records. Bisections give A008438, A062731. Row sums of A127093.

Cf. A144736, A158951, A158902, A174740, A147843.Cf. A001157, A001158, A001160, A001065, A002192, A001001, A001615 (primitive sublattices), A039653, A088580, A074400, A029416, A083728, A006352, A002659, A083238, A000593, A074400, A050449, A050452, A051731, A027748, A124010, A069192.Cf. A009194, A082062 (GCD(a(n),n) and its largest prime factor), A179931, A192795 (GCD(a(n),A001157(n)) and largest prime factor).

Cf. also A034448 (sum of unitary divisors).

Cf. A007955 (products of divisors).

Sequence in context: A097863 A097012 A143348 * A003979 A084250 A090128

Adjacent sequences:  A000200 A000201 A000202 * A000204 A000205 A000206

KEYWORD

easy,core,nonn,nice,mult

AUTHOR

N. J. A. Sloane, Apr 30 1991

STATUS

approved

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Last modified September 19 09:42 EDT 2014. Contains 246975 sequences.