%I A000203 M2329 N0921
%S A000203 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,
%T A000203 40,56,30,72,32,63,48,54,48,91,38,60,56,90,42,96,44,84,78,72,48,124,57,
%U A000203 93,72,98,54,120,72,120,80,90,60,168,62,96,104,127,84,144,68,126,96,144
%N A000203 sigma(n) = sum of divisors of n. Also called sigma_1(n).
%C A000203 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).
%C A000203 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. - comment from Ahmed Fares (ahmedfares(AT)my-deja.com),
Apr 05 2001.
%C A000203 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).
%C A000203 a(n) = number of sublattices of index n in a generic 2-dimensional lattice
- Avi Peretz (njk(AT)netvision.net.il), Jan 29 2001
%C A000203 The sublattices of index n are in one-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.]
%C A000203 Sum of number of common divisors of n and m, where m runs from 1 to n.
- Naohiro Nomoto (pcmusume(AT)m11.alpha-net.ne.jp), Jan 10 2004
%C A000203 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).
%C A000203 Triangle A144736: row sums = sigma(n), right border = phi(n), left border
= d(n). [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Sep 20 2008]
%C A000203 Regarding Euler's recurring sequence for sigma(n). Let s=sigma, then
Euler states [Young, p.361]: "...I say that the value of s(n) can
always be combined from some of the preceding as prescribed by the
following formula: 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) + ..." [From Gary W. Adamson (qntmpkt(AT)yahoo.com),
Oct 05 2008]
%C A000203 Prefaced with a zero: (0, 1, 3, 4, 7,...) = A147843 convolved with the
partition numbers, A000041. [From Gary W. Adamson (qntmpkt(AT)yahoo.com),
Nov 15 2008]
%D A000203 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.
%D A000203 T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag,
1976, page 38.
%D A000203 A. T. Benjamin and J. J. Quinn, Proofs that really count: the art of
combinatorial proof, M.A.A. 2003, p. 116ff.
%D A000203 L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 162, #16, (6), 2nd
formula.
%D A000203 J. W. L. Glaisher, On the function chi(n), Quarterly Journal of Pure
and Applied Mathematics, 20 (1884), 97-167.
%D A000203 M. J. Grady, A group theoretic approach to a famous partition formula,
Amer. Math. Monthly, 112 (No. 7, 2005), 645-651.
%D A000203 Ross Honsberger, "Mathematical Gems, Number One," The Dolciani Mathematical
Expositions, Published and Distributed by The Mathematical Association
of America, page 116.
%D A000203 M. Krasner, Le nombre des surcorps primitifs d'un degre donne et le nombre
des surcorps metagaloisiens d'un degre donne d'un corp de nombre
p-adique. Comptes Redus Hebdomadaires, Acadmie des Science, Paris
254, 255, 1962
%D A000203 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.
%D A000203 P. A. MacMahon, Divisors of numbers and their continuations in the theory
of partitions, Proc. London Math. Soc., 19 (1919), 75-113.
%D A000203 G. Polya, Induction and Analogy in Mathematics, vol. 1 of Mathematics
and Plausible Reasoning, Princeton Univ Press 1954, page 92.
%D A000203 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]
%D A000203 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]
%D A000203 John S. Rutherford, Sublattice enumeration. IV. Equivalence classes of
plane sublattices by parent Patterson symmetry and colour lattice
group type, Acta Cryst. (2009). A65, 156163. [See Table 1]. [From
N. J. A. Sloane, (njas(AT)research.att.com), Feb 23 2009]
%D A000203 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973
(includes this sequence).
%D A000203 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences,
Academic Press, 1995 (includes this sequence).
%D A000203 Robert M. Young, "Excursions in Calculus", The Mathematical Association
of America, 1992 p.361 [From Gary W. Adamson (qntmpkt(AT)yahoo.com),
Oct 05 2008]
%H A000203 Daniel Forgues, <a href="b000203.txt">Table of n, a(n) for n=1..100000</
a>
%H A000203 Walter Nissen, <a href="http://upforthecount.com/math/abundance.html">
Abundancy : Some Resources </a>
%H A000203 M. Abramowitz and I. A. Stegun, eds., <a href="http://www.nrbook.com/
abramowitz_and_stegun/">Handbook of Mathematical Functions</a>, National
Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972
[alternative scanned copy].
%H A000203 M. Baake and U. Grimm, <a href="http://www.ma.utexas.edu/mp_arc-bin/mpa?yn=02-392">
Quasicrystalline combinatorics</a>
%H A000203 H. Bottomley, <a href="a203.gif">Illustration of initial terms</a>
%H A000203 C. K. Caldwell, The Prime Glossary, <a href="http://primes.utm.edu/glossary/
page.php?sort=SigmaFunction">sigma function</a>
%H A000203 L. Euler, <a href="http://math.dartmouth.edu/~euler/pages/E243.html">
Observatio de summis divisorum</a>
%H A000203 L. Euler, <a href="http://arXiv.org/abs/math.HO/0411587">An observation
on the sums of divisors</a>
%H A000203 Daniele A. Gewurz and Francesca Merola, <a href="http://www.cs.uwaterloo.ca/
journals/JIS/index.html">Sequences realized as Parker vectors ...</
a>, J. Integer Seqs., Vol. 6, 2003.
%H A000203 M. Maia and M. Mendez, <a href="http://arXiv.org/abs/math.CO/0503436">
On the arithmetic product of combinatorial species</a>
%H A000203 K. Matthews, <a href="http://www.numbertheory.org/php/factor.html">Factorizing
n and calculating phi(n),omega(n),d(n),sigma(n) and mu(n)</a>
%H A000203 Jon Perry, <a href="http://www.users.globalnet.co.uk/~perry/maths/morepartitionfunction/
morepartitionfunction.htm">More Partition Functions</a>
%H A000203 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
DivisorFunction.html">Link to a section of The World of Mathematics.</
a>
%H A000203 <a href="Sindx_Su.html#sublatts">Index entries for sequences related
to sublattices</a>
%H A000203 <a href="Sindx_Cor.html#core">Index entries for "core" sequences</a>
%F A000203 Dirichlet convolution of phi(n) and tau(n), i.e. a(n)=Sum_{d|n} phi(n/
d)*tau(d), cf. A000010, A000005.
%F A000203 Multiplicative with a(p^e) = (p^(e+1)-1)/(p-1). - David W. Wilson (davidwwilson(AT)comcast.net),
Aug 01, 2001.
%F A000203 sigma(n) is odd iff n is a square or twice a square. - Robert G. Wilson
v (rgwv(AT)rgwv.com), Oct 03 2001
%F A000203 sigma[n]=sigma[n*p(n)]-p(n)*sigma[n] - Labos E. (labos(AT)ana.sote.hu),
Aug 14 2003
%F A000203 a(n) = n*A000041(n) - sum{i=1, n-1, a(i)*A000041(n-i)} - Jon Perry (perry(AT)globalnet.co.uk),
Sep 11 2003
%F A000203 a(n) = -A010815(n)*n - Sum(A010815(k)*a(n-k): 1<=k<n). - Reinhard Zumkeller
(reinhard.zumkeller(AT)gmail.com), Nov 30 2003
%F A000203 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 (reinhard.zumkeller(AT)gmail.com),
Nov 17 2004
%F A000203 Recurrence: sigma(1) = 1 sigma(n) = 12*Sum[(5*k*(n-k)-n^2)*sigma(k)*sigma(n-k),
k=1..(n-1)]/((n^2)*(n-1)) if n>1 - Dominique Giard (dominique.giard(AT)gmail.com),
Jan 11 2005
%F A000203 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
%F A000203 For odd n, A000203(n) = A000593(n) sum of odd divisors of n. For even
n, A000203(n) = A000593(n) + A074400(n/2) where A074400 is sum of
the even divisors of 2n. - Jonathan Vos Post (jvospost3(AT)gmail.com),
Mar 26 2006
%F A000203 Equals A051731 * [1,2,3,...]; the inverse Moebius transform of the natural
numbers. Equals row sums of A127093 - Gary W. Adamson (qntmpkt(AT)yahoo.com),
May 20 2007
%F A000203 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 (qntmpkt(AT)yahoo.com),
Oct 31 2007
%F A000203 Row sums of triangle A134838 - Gary W. Adamson (qntmpkt(AT)yahoo.com),
Nov 12 2007
%F A000203 a(n) = A054785(2*n) - A000593(2*n). - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com),
Apr 23 2008
%e A000203 For example, 6 is divisible by 1, 2, 3 and 6, so sigma(6) = 1 + 2 + 3
+ 6 = 12.
%e A000203 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.
%p A000203 with(numtheory): A000203 := n->sigma(n);
%t A000203 Table[ DivisorSigma[1, n], {n, 1, 100} ]
%o A000203 (MAGMA) [ SumOfDivisors(n) : n in [1..40]];
%o A000203 (PARI) a(n)=if(n<1,0,sigma(n))
%o A000203 (PARI) a(n)=if(n<1,0,direuler(p=2,n,1/(1-X)/(1-p*X))[n])
%o A000203 (PARI) a(n)=if(n<1,0,polcoeff(sum(k=1,n,x^k/(x^k-1)^2,x*O(x^n)),n)) /
* Michael Somos Jan 29 2005 */
%o A000203 (MuPad) numlib::sigma(n)$ n=1..81 - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com),
May 13 2008
%o A000203 (PARI from Joerg Arndt (arndt(AT)jjj.de), May 03, 2008)
%o A000203 N=17; default(seriesprecision,N); x=z+O(z^(N+1))
%o A000203 c=sum(j=1,N,j*x^j); \\ log case
%o A000203 s=-log(prod(j=1,N,1-x^j)); \\ A000203 sum of divisors
%o A000203 s=serconvol(s,c)
%o A000203 v=Vec(s)
%o A000203 (Other) sage: [sigma(n,1)for n in xrange(1,71)] # [From Zerinvary Lajos
(zerinvarylajos(AT)yahoo.com), Jun 04 2009]
%o A000203 (PARI) max_n = 30; ser = - sum(k=1,max_n,log(1-x^k)) a(n) = polcoeff(ser,
n)*n [From Gottfried Helms (helms(AT)uni-kassel.de), Aug 10 2009]
%Y A000203 Cf. A001157, A001158, A001160, A001065, A002192, A001001, A001615 (primitive
sublattices).
%Y A000203 See A034885, A002093 for records. Bisections give A008438, A062731.
%Y A000203 Cf. A039653, A088580, A074400, A029416, A083728, A006352, A002659, A083238.
%Y A000203 Cf. A000593, A074400, A050449, A050452.
%Y A000203 Cf. A051731, A127093.
%Y A000203 Cf. A134838.
%Y A000203 A144736 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Sep 20 2008]
%Y A000203 A147843 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Nov 15 2008]
%Y A000203 Equals row sums of triangle A158951 [From Gary W. Adamson (qntmpkt(AT)yahoo.com),
Mar 31 2009]
%Y A000203 Equals row sums of triangle A158902 [From Gary W. Adamson (qntmpkt(AT)yahoo.com),
Mar 29 2009]
%Y A000203 Sequence in context: A097863 A097012 A143348 this_sequence A003979 A084250
A090128
%Y A000203 Adjacent sequences: A000200 A000201 A000202 this_sequence A000204 A000205
A000206
%K A000203 easy,core,nonn,nice,mult
%O A000203 1,2
%A A000203 N. J. A. Sloane (njas(AT)research.att.com).
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