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A065764
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Sum of divisors of square numbers.
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49
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1, 7, 13, 31, 31, 91, 57, 127, 121, 217, 133, 403, 183, 399, 403, 511, 307, 847, 381, 961, 741, 931, 553, 1651, 781, 1281, 1093, 1767, 871, 2821, 993, 2047, 1729, 2149, 1767, 3751, 1407, 2667, 2379, 3937, 1723, 5187, 1893, 4123, 3751, 3871, 2257, 6643
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OFFSET
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1,2
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COMMENTS
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Unlike A065765, the sums of divisors of squares give remainders r=1,3,5 modulo 6: sigma(4)==1, sigma(49)==3, sigma(2401)==5 (mod 6). See also A097022.
a(n) is also the number of ordered pairs of positive integers whose LCM is n, (see LeVeque). - Enrique Pérez Herrero, Aug 26 2013
Main diagonal of A319526. - Omar E. Pol, Sep 25 2018
Subsequence of primes is A023195 \ {3}; also, 31 is the only known prime to be twice in the data because 31 = sigma(16) = sigma(25) (see A119598 and Goormaghtigh conjecture link). - Bernard Schott, Jan 17 2021
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REFERENCES
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W. J. LeVeque, Fundamentals of Number Theory, pp. 125 Problem 4, Dover NY 1996.
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LINKS
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T. D. Noe, Table of n, a(n) for n=1..10000
Wikipedia, Goormaghtigh conjecture.
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FORMULA
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a(n) = sigma(n^2) = A000203(A000290(n)).
Multiplicative with a(p^e) = (p^(2*e+1)-1)/(p-1). - Vladeta Jovovic, Dec 01 2001
Dirichlet g.f.: zeta(s)*zeta(s-1)*zeta(s-2)/zeta(2*s-2), inverse Mobius transform of A000082. - R. J. Mathar, Mar 06 2011
Dirichlet convolution of A001157 by the absolute terms of A055615. Also the Dirichlet convolution of A048250 by A000290. - R. J. Mathar, Apr 12 2011
a(n) = Sum_{d|n} d*Psi(d), where Psi is A001615. - Enrique Pérez Herrero, Feb 25 2012
a(n) >= (n+1) * sigma(n) - n, where sigma is A000203, equality holds if n is in A000961. - Enrique Pérez Herrero, Apr 21 2012
Sum_{k=1..n} a(k) ~ 5*Zeta(3) * n^3 / Pi^2. - Vaclav Kotesovec, Jan 30 2019
Sum_{k>=1} 1/a(k) = 1.3947708738535614499846243600124612760835313454790187655653356563282177118... - Vaclav Kotesovec, Sep 20 2020
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MAPLE
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with(numtheory): [sigma(n^2)$n=1..50]; # Muniru A Asiru, Jan 01 2019
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MATHEMATICA
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Table[Plus@@Divisors[n^2], {n, 48}] (* Alonso del Arte, Feb 24 2012 *)
f[p_, e_] := (p^(2*e + 1) - 1)/(p - 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 50] (* Amiram Eldar, Sep 10 2020 *)
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PROG
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(MuPAD) numlib::sigma(n^2)$ n=1..81 // Zerinvary Lajos, May 13 2008
(Sage) [sigma(n^2, 1)for n in range(1, 49)] # Zerinvary Lajos, Jun 13 2009
(PARI) a(n) = sigma(n^2); \\ Harry J. Smith, Oct 30 2009
(Magma) [SumOfDivisors(n^2): n in [1..48]]; // Bruno Berselli, Apr 12 2011
(GAP) a:=List([1..50], n->Sigma(n^2));; Print(a); # Muniru A Asiru, Jan 01 2019
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CROSSREFS
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Cf. A000203, A000290, A028982, A319526.
Sequence in context: A283709 A063583 A350072 * A273757 A073473 A272407
Adjacent sequences: A065761 A065762 A065763 * A065765 A065766 A065767
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KEYWORD
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nonn,easy,mult
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AUTHOR
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Labos Elemer, Nov 19 2001
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STATUS
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approved
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