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A001157 sigma_2(n): sum of squares of divisors of n.
(Formerly M3799 N1551)
237
1, 5, 10, 21, 26, 50, 50, 85, 91, 130, 122, 210, 170, 250, 260, 341, 290, 455, 362, 546, 500, 610, 530, 850, 651, 850, 820, 1050, 842, 1300, 962, 1365, 1220, 1450, 1300, 1911, 1370, 1810, 1700, 2210, 1682, 2500, 1850, 2562, 2366, 2650, 2210, 3410, 2451, 3255 (list; graph; refs; listen; history; text; internal format)
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.

Row sums of triangles A134575 and A134559. - Gary W. Adamson, Nov 02 2007

Also sum of square divisors of n^2. - Michel Marcus, Jan 14 2014

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).

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

Index entries for "core" sequences

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(prod(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

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

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 xrange(1, 51)] # [From 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

CROSSREFS

Cf. A000005, A000203, A001158, A001159, A053807, A064602, A127093, A134841.

Cf. A192794, A082063 (GCD(a(n),n) and its largest prime factor); A179931, A192795 (GCD(a(n),A000203(n)) and largest prime factor).

Main diagonal of the array in A242639.

Sequence in context: A017667 A241603 A242643 * A242644 A002800 A132174

Adjacent sequences:  A001154 A001155 A001156 * A001158 A001159 A001160

KEYWORD

nonn,core,nice,easy,mult

AUTHOR

N. J. A. Sloane, R. K. Guy

EXTENSIONS

More terms from Stefan Steinerberger, Mar 24 2006

STATUS

approved

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Last modified November 28 07:52 EST 2014. Contains 250285 sequences.