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A001157
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sigma_2(n): sum of squares of divisors of n.
(Formerly M3799 N1551)
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226
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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
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OFFSET
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1,2
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COMMENTS
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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
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REFERENCES
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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.
Andrews, George E., Stacked lattice boxes, Ann. Comb. 3 (1999), 115-130. See (2.3).
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.
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).
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LINKS
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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].
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
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FORMULA
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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
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MAPLE
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with(numtheory); A001157 := n->sigma[2](n); [seq(sigma[2](n), n=1..100)];
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MATHEMATICA
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Table[DivisorSigma[2, n], {n, 1, 50}] - Stefan Steinerberger, Mar 24 2006
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PROG
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(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 4 2011 */
(Sage) [sigma(n, 2)for n in xrange(1, 51)] # [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), 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
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CROSSREFS
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Cf. A000005, A000203, A001158, A001159.
Cf. A053807, A064602.
Cf. A127093.
Cf. A134841.
Cf. A192794, A082063 (GCD(a(n),n) and its largest prime factor); A179931, A192795 (GCD(a(n),A000203(n)) and largest prime factor).
Sequence in context: A002791 A080399 A017667 * A002800 A132174 A132461
Adjacent sequences: A001154 A001155 A001156 * A001158 A001159 A001160
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KEYWORD
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nonn,core,nice,easy,mult
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AUTHOR
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N. J. A. Sloane, R. K. Guy
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EXTENSIONS
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More terms from Stefan Steinerberger, Mar 24 2006
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STATUS
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approved
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