(Greetings from !)
a(n) = sigma(n), the sum of the divisors of n. Also called sigma_1(n).
(Formerly M)
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
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 - 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
(this sequence) (k=1), - (k=2,3,4,5), - for k = 6,7,...,24. - Ahmed Fares (ahmedfares(AT)), Apr 05 2001
A number n is abundant if sigma(n) & 2n (cf. ), perfect if sigma(n) = 2n (cf. ), deficient if sigma(n) & 2n (cf. ).
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 [ 0 d] with a&0, ad=n, b in [0..d-1]. The number of these is Sum_{d|n} d = sigma(n), which is a(n). A sublattice is primitive if gcd(a,b,d) = 1; the number of these is n * product_{p|n} (1+1/p), which is . [Cf. Grady reference.]
Sum of number of common divisors of n and m, where m runs from 1 to n. - , 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. , ,
(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. - , Oct 05 2008 [corrected Apr 27 2012 by
based on Ewell]
Prefaced with a zero: (0, 1, 3, 4, 7,...) =
convolved with the partition numbers, . - , 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. - , Nov 08 2012
Also total number of parts in the partitions of n into equal parts. - , 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. - , Dec 21 2013, based on postings to the Number Theory Mailing List by
a(n) = (n, (n)). - , Apr 10 2014
lim_{m-&infinity} (Sum_{n=1..prime(m)} a(n)) / prime(m)^2 = zeta(2)/2. See more at . - , Jan 04 2015
a(n) + (n) is an odd number iff n = 2m^2, m&=1. - , Jan 15 2015
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.
P. Pollack, C. Pomerance, Some problems of Erdos on the sum-of-divisors function, For Richard Guy on his 99th birthday: May his sequence be unbounded, 2015, http://alpha.math.uga.edu/~pollack/reversal12.pdf
G. Polya, Induction and Analogy in Mathematics, vol. 1 of Mathematics and Plausible Reasoning, Princeton Univ Press 1954, page 92.
J. S. Rutherford, The enumeration and symmetry-significant properties of derivative lattices, Acta Cryst. A48 (1992), 500-508. [From , Mar 14 2009]
J. S. Rutherford, The enumeration and symmetry-significant properties of derivative lattices II, Acta Cryst. A49 (1993), 293-300. [From , 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 , Oct 05 2008]
N. J. A. Sloane and Daniel Forgues,
(first 20000 terms from N. J. A. Sloane)
M. Abramowitz and I. A. Stegun, eds., , National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
B. Apostol, L. Petrescu, , Journal of Integer Sequences, 2013, # 13.7.5.
M. Baake and U. Grimm,
H. Bottomley,
C. K. Caldwell, The Prime Glossary,
J. N. Cooper and A. W. N. Riasanovsky, , 2012. - From N. J. A. Sloane, Dec 25 2012
L. Euler, , arXiv:math/0411587 [math.HO], .
J. A. Ewell, , Proc. Amer. Math. Soc. 64 (2) 1977.
Daniele A. Gewurz and Francesca Merola, , J. Integer Seqs., Vol. 6, 2003.
J. W. L. Glaisher, , Quarterly Journal of Pure and Applied Mathematics, 20 (1884), 97-167. [Annotated scanned copy]
M. J. Grady, , Amer. Math. Monthly, 112 (No. 7, 2005), 645-651.
P. A. MacMahon, , Proc. London Math. Soc., 19 (1921), 75-113.
M. Maia and M. Mendez, , arXiv:math.CO/0503436
K. Matthews,
Walter Nissen,
Carl Pomerance and Hee-Sung Yang, , Math. Comp. 83 (2014), .
John S. Rutherford, , Acta Cryst. (2009). A65, 156-163. [See Table 1]. - From , Feb 23 2009
Eric Weisstein's World of Mathematics,
Wikipedia,
Multiplicative with a(p^e) = (p^(e+1)-1)/(p-1). - , Aug 01 2001
G.f.: -x*deriv(eta(x))/eta(x) where eta(x) = Product_{n&=1} (1-x^n). - , Mar 14 2010
L.g.f.: -log(Product_{j&=1} (1-x^j)) = Sum_{n&=1} a(n)/n*x^n. - , Feb 04 2011
Dirichlet convolution of phi(n) and tau(n), i.e., a(n) = sum_{d|n} phi(n/d)*tau(d), cf. , .
a(n) is odd iff n is a square or twice a square. - , Oct 03 2001
a(n) = a(n*p(n)) - p(n)*a(n). - , Aug 14 2003
a(n) = n*(n) - Sum_{i=1..n-1} a(i)*(n-i). - , Sep 11 2003
a(n) = -(n)*n - Sum_{k=1..n-1} (k)*a(n-k). - , 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 (). - , Nov 17 2004
Recurrence: n^2*(n-1)*a(n) = 12*Sum_{k=1..n-1} (5*k*(n-k) - n^2)*a(k)*a(n-k), if n&1. - Dominique Giard (dominique.giard(), 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). - , Apr 05 2003
For odd n, a(n) = (n) sum of odd divisors of n. For even n, a(n) = (n) + (n/2) where
is sum of the even divisors of 2n. - , Mar 26 2006
Equals the inverse Moebius transform of the natural numbers. Equals row sums of . - , May 20 2007
* [1/1, 1/2, 1/3,...] = [1/1, 3/2, 4/3, 7/4, 6/5, 12/6, 8/7,...]. Row sums of triangle . - , Oct 31 2007
Row sums of triangle . - , Nov 12 2007
a(n) = (2*n) - (2*n). - , Apr 23 2008
a(n) = n*Sum_{k=1..n} (n,k)/k*(-1)^(k+1). - , Aug 10 2010
Dirichlet convolution of
and . - , 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). - , Dec 06 2011
a(n) = (n) + n. - , May 20 2012
a(n) = (n) - (n). - , Jan 17 2013
a(n) = Sum_{k=1..(n)} (-1)^(k-1)*(n,k). - conjectured by , Feb 02 2013, and proved by , Nov 17 2013
a(n) = Sum_{k=1..(n)} (-1)^(k-1)*(k)*(n-(k)). - , Mar 05 2014
It appears that a(n) & 6*n^(3/2)/Pi^2 for n & 12. - , May 14 2014
a(n) = Sum_{d^2|n} (n/d^2) = Sum_{d^3|n} (n/d^3). - , Mar 06 2015
a(3*n) = (n). a(3*n + 1) = (n). a(3*n + 2) = (n). - , Jul 19 2015
a(n) = Sum{i=1..n} Sum{j=1..i} cos((2*Pi*n*j)/i). - , Oct 14 2015
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 .
with(numtheory):
:= n-&sigma(n); seq((n), n=1..100);
MATHEMATICA
Table[ DivisorSigma[1, n], {n, 100}]
a[ n_] := SeriesCoefficient[ QPolyGamma[ 1, 1, q] / Log[q]^2, {q, 0, n}]; (* , Apr 25 2013 *)
(MAGMA) [SumOfDivisors(n): n in [1..70]];
(MAGMA) [DivisorSigma(1, n): n in [1..70]]; // , Sep 09 2015
(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))}; /* , Jan 29 2005 */
(MuPad) numlib::sigma(n)$ n=1..81 // , May 13 2008
(Sage) [sigma(n, 1) for n in xrange(1, 71)] # , Jun 04 2009
(PARI) max_n = 30; ser = - sum(k=1, max_n, log(1-x^k)) a(n) = polcoeff(ser, n)*n \\ , Aug 10 2009
(Maxima) makelist(divsum(n), n, 1, 1000); \\ , Mar 26 2011
a000203 n = product $ zipWith (\p e -& (p^(e+1)-1) `div` (p-1)) (a027748_row n) (a124010_row n)
-- , May 07 2012
(MIT/GNU Scheme, with Aubrey Jaffer's SLIB Scheme library)
(require 'factor)
(define ( 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))))))
;; , Dec 02 2013
for records. Bisections give , . Row sums of .
Cf. , , , , , , , , , , , ,
(primitive sublattices), , , , , , , , , , , , , , , , .Cf. ,
(GCD(a(n),n) and its largest prime factor), ,
(GCD(a(n),(n)) and largest prime factor).
(sum of unitary divisors).
(products of divisors).
Sequence in context:
Adjacent sequences:&&
easy,core,nonn,nice,mult
, Apr 30 1991
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Last modified November 13 23:01 EST 2015.
Contains 263885 sequences.