OFFSET
1,2
COMMENTS
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).
sigma_2(n) is the sum of the squares of the divisors of n.
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 (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.
Also sum of square divisors of n^2. - Michel Marcus, Jan 14 2014
Conjecture: For each k = 2,3,..., all the rational numbers sigma_k(n)/n^k = Sum_{d|n} 1/d^k (n = 1,2,3,...) have pairwise distinct fractional parts. - Zhi-Wei Sun, Oct 15 2015
5 is the only prime entry in the sequence. - Drake Thomas, Dec 18 2016
4*a(n) = sum of squares of even divisors of 2*n. - Wolfdieter Lang, Jan 07 2017
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. 827.
T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, page 38.
D. M. Bressoud, Proofs and Confirmations, Camb. Univ. Press, 1999; p. 11.
P. A. MacMahon, The connexion between the sum of the squares of the divisors and the number of partitions of a given number, Messenger Math., 54 (1924), 113-116. Collected Papers, MIT Press, 1978, Vol. I, pp. 1364-1367. See Table I. The entry 53 should be 50. - N. J. A. Sloane, May 21 2014
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].
George E. Andrews, Stacked lattice boxes, Ann. Comb. 3 (1999), 115-130. See (2.3).
Joerg Arndt, On computing the generalized Lambert series, arXiv:1202.6525v3 [math.CA], (2012).
Y. Puri and T. Ward, Arithmetic and growth of periodic orbits, J. Integer Seqs., Vol. 4 (2001), #01.2.1.
Eric Weisstein's World of Mathematics, Divisor Function
FORMULA
G.f.: Sum_{k>0} k^2 x^k/(1-x^k). Dirichlet g.f.: zeta(s)*zeta(s-2). - Michael Somos, Apr 05 2003
Multiplicative with a(p^e) = (p^(2e+2)-1)/(p^2-1). - David W. Wilson, Aug 01 2001
G.f. for sigma_k(n): Sum_{m>0} m^k*x^m/(1-x^m). - Vladeta Jovovic, Oct 18 2002
L.g.f.: -log(Product_{j>=1} (1-x^j)^j) = Sum_{n>=1} a(n)/n*x^n. - Joerg Arndt, Feb 04 2011
Equals A127093 * [1, 2, 3, ...]. - Gary W. Adamson, May 10 2007
Equals A051731 * [1, 4, 9, 16, 25, ...]. A051731 * [1/1, 1/2, 1/3, 1/4, ...] = [1/1, 5/4, 10/9, 21/16, 26/25, ...]. - Gary W. Adamson, Nov 02 2007
Row sums of triangle A134841. - Gary W. Adamson, Nov 12 2007
a(n) = A035316(n^2). - Michel Marcus, Jan 14 2014
Conjecture: a(n) = sigma(n^2*rad(n))/sigma(rad(n)), where sigma = A000203 and rad = A007947. - Velin Yanev, Aug 20 2017
G.f.: Sum_{k>=1} x^k*(1 + x^k)/(1 - x^k)^3. - Ilya Gutkovskiy, Oct 24 2018
a(n) = a(n/4) + A050461(n) + A076577(n/2) + A050465(n) where A(.) are zero for non-integer arguments. - R. J. Mathar, May 25 2020
Sum_{k>=1} 1/a(k) = A109694 = 1.53781289182725616253866100273826833091936004947322354929617689659426330445... - Vaclav Kotesovec, Sep 26 2020
G.f.: Sum_{n >= 1} q^(n^2)*(n^2 - ((n-1)^2 - 2)*q^n - ((n+1)^2 - 2)*q^(2*n) + n^2*q^(3*n))/(1 - q^n)^3 - apply the operator x*d/dx twice to equation 5 in Arndt and set x = 1. - Peter Bala, Jan 21 2021
From Vaclav Kotesovec, Aug 07 2022: (Start)
Sum_{k=1..n} a(k) = A064602(n) ~ zeta(3) * n^3 / 3.
Sum_{k=1..n} (-1)^k * a(k) ~ zeta(3) * n^3 / 24. (End)
a(n) = Sum_{1 <= i, j <= n} tau(gcd(i, j, n)) = Sum_{d divides n} tau(d) * J_2(n/d), where the divisor function tau(n) = A000005(n) and the Jordan totient function J_2(n) = A007434(n). - Peter Bala, Jan 22 2024
MAPLE
with(numtheory); A001157 := n->sigma[2](n); [seq(sigma[2](n), n=1..100)];
MATHEMATICA
Table[DivisorSigma[2, n], {n, 1, 50}] (* Stefan Steinerberger, Mar 24 2006 *)
DivisorSigma[2, Range[50]] (* Harvey P. Dale, Aug 22 2016 *)
PROG
(PARI) a(n)=if(n<1, 0, sigma(n, 2))
(PARI) a(n)=if(n<1, 0, direuler(p=2, n, 1/(1-X)/(1-p^2*X))[n])
(PARI) a(n)=if(n<1, 0, n*polcoeff(sum(k=1, n, x^k/(x^k-1)^2/k, x*O(x^n)), n)) /* Michael Somos, Jan 29 2005 */
(PARI) N=99; q='q+O('q^N); Vec(sum(n=1, N, n^2*q^n/(1-q^n))) /* Joerg Arndt, Feb 04 2011 */
(PARI) a(n) = sumdiv(n^2, d, issquare(d)*d); \\ Michel Marcus, Jan 14 2014
(Sage) [sigma(n, 2)for n in range(1, 51)] # Zerinvary Lajos, Jun 04 2009
(Maxima) makelist(divsum(n, 2), n, 1, 20); /* Emanuele Munarini, Mar 26 2011 */
(Haskell)
a001157 n = s n 1 1 a000040_list where
s 1 1 y _ = y
s m x y ps'@(p:ps)
| m `mod` p == 0 = s (m `div` p) (x * p^2) y ps'
| x > 1 = s m 1 (y * (x * p^2 - 1) `div` (p^2 - 1)) ps
| otherwise = s m 1 y ps
-- Reinhard Zumkeller, Jul 10 2011
(Magma) [DivisorSigma(2, n): n in [1..50]]; // Bruno Berselli, Apr 10 2013
(Python)
from sympy import divisor_sigma
def a(n): return divisor_sigma(n, 2)
print([a(n) for n in range(1, 51)]) # Michael S. Branicky, Jan 05 2021
(Python)
from math import prod
from sympy import factorint
def a(n): return prod((p**(2*e+2)-1)//(p**2-1) for p, e in factorint(n).items())
print([a(n) for n in range(1, 51)]) # Michael S. Branicky, Feb 25 2024
CROSSREFS
KEYWORD
nonn,core,nice,easy,mult,changed
AUTHOR
STATUS
approved