Search: id:A000203 Results 1-1 of 1 results found. %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, Table of n, a(n) for n=1..100000 %H A000203 Walter Nissen, Abundancy : Some Resources %H A000203 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]. %H A000203 M. Baake and U. Grimm, Quasicrystalline combinatorics %H A000203 H. Bottomley, Illustration of initial terms %H A000203 C. K. Caldwell, The Prime Glossary, sigma function %H A000203 L. Euler, Observatio de summis divisorum %H A000203 L. Euler, An observation on the sums of divisors %H A000203 Daniele A. Gewurz and Francesca Merola, Sequences realized as Parker vectors ..., J. Integer Seqs., Vol. 6, 2003. %H A000203 M. Maia and M. Mendez, On the arithmetic product of combinatorial species %H A000203 K. Matthews, Factorizing n and calculating phi(n),omega(n),d(n),sigma(n) and mu(n) %H A000203 Jon Perry, More Partition Functions %H A000203 Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics. %H A000203 Index entries for sequences related to sublattices %H A000203 Index entries for "core" sequences %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<=k1 - 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 = be a 2-dimensional lattice. The 7 sublattices of index 4 are generated by <4V,W>, , <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). Search completed in 0.007 seconds