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%I M3799 N1551 #134 Feb 25 2024 10:13:40
%S 1,5,10,21,26,50,50,85,91,130,122,210,170,250,260,341,290,455,362,546,
%T 500,610,530,850,651,850,820,1050,842,1300,962,1365,1220,1450,1300,
%U 1911,1370,1810,1700,2210,1682,2500,1850,2562,2366,2650,2210,3410,2451,3255
%N a(n) = sigma_2(n): sum of squares of divisors of n.
%C 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 sigma_2(n) is the sum of the squares of the divisors of n.
%C 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.
%C Row sums of triangles A134575 and A134559. - _Gary W. Adamson_, Nov 02 2007
%C Also sum of square divisors of n^2. - _Michel Marcus_, Jan 14 2014
%C 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
%C 5 is the only prime entry in the sequence. - _Drake Thomas_, Dec 18 2016
%C 4*a(n) = sum of squares of even divisors of 2*n. - _Wolfdieter Lang_, Jan 07 2017
%D 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.
%D T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, page 38.
%D D. M. Bressoud, Proofs and Confirmations, Camb. Univ. Press, 1999; p. 11.
%D 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
%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H T. D. Noe, <a href="/A001157/b001157.txt">Table of n, a(n) for n = 1..10000</a>
%H M. Abramowitz and I. A. Stegun, eds., <a href="http://www.convertit.com/Go/ConvertIt/Reference/AMS55.ASP">Handbook of Mathematical Functions</a>, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
%H George E. Andrews, <a href="http://dx.doi.org/10.1007/BF01608779">Stacked lattice boxes</a>, Ann. Comb. 3 (1999), 115-130. See (2.3).
%H Joerg Arndt, <a href="http://arxiv.org/abs/1202.6525">On computing the generalized Lambert series</a>, arXiv:1202.6525v3 [math.CA], (2012).
%H Y. Puri and T. Ward, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL4/WARD/short.html">Arithmetic and growth of periodic orbits</a>, J. Integer Seqs., Vol. 4 (2001), #01.2.1.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/DivisorFunction.html">Divisor Function</a>
%H <a href="/index/Si#SIGMAN">Index entries for sequences related to sigma(n)</a>
%H <a href="/index/Cor#core">Index entries for "core" sequences</a>
%F 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
%F Multiplicative with a(p^e) = (p^(2e+2)-1)/(p^2-1). - _David W. Wilson_, Aug 01 2001
%F G.f. for sigma_k(n): Sum_{m>0} m^k*x^m/(1-x^m). - _Vladeta Jovovic_, Oct 18 2002
%F L.g.f.: -log(Product_{j>=1} (1-x^j)^j) = Sum_{n>=1} a(n)/n*x^n. - _Joerg Arndt_, Feb 04 2011
%F Equals A127093 * [1, 2, 3, ...]. - _Gary W. Adamson_, May 10 2007
%F 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
%F Row sums of triangle A134841. - _Gary W. Adamson_, Nov 12 2007
%F a(n) = A035316(n^2). - _Michel Marcus_, Jan 14 2014
%F Conjecture: a(n) = sigma(n^2*rad(n))/sigma(rad(n)), where sigma = A000203 and rad = A007947. - _Velin Yanev_, Aug 20 2017
%F G.f.: Sum_{k>=1} x^k*(1 + x^k)/(1 - x^k)^3. - _Ilya Gutkovskiy_, Oct 24 2018
%F 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
%F Sum_{k>=1} 1/a(k) = A109694 = 1.53781289182725616253866100273826833091936004947322354929617689659426330445... - _Vaclav Kotesovec_, Sep 26 2020
%F 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
%F From _Vaclav Kotesovec_, Aug 07 2022: (Start)
%F Sum_{k=1..n} a(k) = A064602(n) ~ zeta(3) * n^3 / 3.
%F Sum_{k=1..n} (-1)^k * a(k) ~ zeta(3) * n^3 / 24. (End)
%F 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
%p with(numtheory); A001157 := n->sigma[2](n); [seq(sigma[2](n), n=1..100)];
%t Table[DivisorSigma[2, n], {n, 1, 50}] (* _Stefan Steinerberger_, Mar 24 2006 *)
%t DivisorSigma[2,Range[50]] (* _Harvey P. Dale_, Aug 22 2016 *)
%o (PARI) a(n)=if(n<1,0,sigma(n,2))
%o (PARI) a(n)=if(n<1,0,direuler(p=2,n,1/(1-X)/(1-p^2*X))[n])
%o (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 */
%o (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 */
%o (PARI) a(n) = sumdiv(n^2, d, issquare(d)*d); \\ _Michel Marcus_, Jan 14 2014
%o (Sage) [sigma(n,2)for n in range(1,51)] # _Zerinvary Lajos_, Jun 04 2009
%o (Maxima) makelist(divsum(n,2),n,1,20); \\ _Emanuele Munarini_, Mar 26 2011
%o (Haskell)
%o a001157 n = s n 1 1 a000040_list where
%o s 1 1 y _ = y
%o s m x y ps'@(p:ps)
%o | m `mod` p == 0 = s (m `div` p) (x * p^2) y ps'
%o | x > 1 = s m 1 (y * (x * p^2 - 1) `div` (p^2 - 1)) ps
%o | otherwise = s m 1 y ps
%o -- _Reinhard Zumkeller_, Jul 10 2011
%o (Magma) [DivisorSigma(2,n): n in [1..50]]; // _Bruno Berselli_, Apr 10 2013
%o (Python)
%o from sympy import divisor_sigma
%o def a(n): return divisor_sigma(n, 2)
%o print([a(n) for n in range(1, 51)]) # _Michael S. Branicky_, Jan 05 2021
%o (Python)
%o from math import prod
%o from sympy import factorint
%o def a(n): return prod((p**(2*e+2)-1)//(p**2-1) for p, e in factorint(n).items())
%o print([a(n) for n in range(1, 51)]) # _Michael S. Branicky_, Feb 25 2024
%Y Cf. A000005, A000203, A001158, A001159, A053807, A064602, A127093, A134841, A048250.
%Y Cf. also A192794, A082063 (gcd(a(n),n) and its largest prime factor); A179931, A192795 (gcd(a(n),A000203(n)) and largest prime factor).
%Y Main diagonal of the array in A242639.
%Y Cf. A333972 (Dgf at s=4).
%K nonn,core,nice,easy,mult
%O 1,2
%A _N. J. A. Sloane_, _R. K. Guy_
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