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A060872 Sum of d*d' over all unordered pairs (d,d') with d*d' = n.

%I A060872 #41 Oct 21 2024 05:36:12
%S A060872 1,2,3,8,5,12,7,16,18,20,11,36,13,28,30,48,17,54,19,60,42,44,23,96,50,
%T A060872 52,54,84,29,120,31,96,66,68,70,180,37,76,78,160,41,168,43,132,135,92,
%U A060872 47,240,98,150,102,156,53,216,110,224,114,116,59,360,61,124,189,256
%N A060872 Sum of d*d' over all unordered pairs (d,d') with d*d' = n.
%C A060872 a(n) is also the sum of all parts of all partitions of n into consecutive parts that differ by 2. - _Omar E. Pol_, May 05 2020
%H A060872 Vincenzo Librandi, <a href="/A060872/b060872.txt">Table of n, a(n) for n = 1..5000</a>
%F A060872 a(n) = n * ceiling( d(n)/2) where d is the number of divisors function.
%F A060872 G.f.: x*f'(x), where f(x) = Sum_{k>=1} x^k^2/(1 - x^k). - _Ilya Gutkovskiy_, Apr 10 2017
%F A060872 a(n) = n*A038548(n). - _Omar E. Pol_, May 05 2020
%F A060872 Dirichlet g.f.: (zeta(2*s-2) + zeta(s-1)^2)/2. - _Vaclav Kotesovec_, Oct 21 2024
%e A060872 a(4)=8 because pairs of factors are 1*4 and 2*2 and 1*4 + 2*2 = 8.
%e A060872 For n = 16 there are three partitions of 16 into consecutive parts that differ by 2 (including 16 as a valid partition). They are [16], [9, 7] and [7, 5, 3, 1]. The sum of the parts is [16] + [9 + 7] + [7 + 5 + 3 + 1] = 48, so a(16) = 48. - _Omar E. Pol_, May 05 2020
%t A060872 Table[ n * Ceiling[ DivisorSigma[0, n] /2 ], {n, 1, 73} ]
%o A060872 (Magma) [n*Ceiling(DivisorSigma(0, n)/2): n in [1..70]]; // _Vincenzo Librandi_, Apr 12 2017
%o A060872 (Python)
%o A060872 from sympy import divisor_count
%o A060872 def A060872(n): return n*(divisor_count(n)+1>>1) # _Chai Wah Wu_, Jul 11 2023
%o A060872 (PARI) a(n) = n*ceil(numdiv(n)/2); \\ _Michel Marcus_, Jul 12 2023
%Y A060872 Cf. A038548, A060866.
%Y A060872 First differences of A083356.
%K A060872 nonn,easy
%O A060872 1,2
%A A060872 _N. J. A. Sloane_, May 04 2001
%E A060872 More terms from _Robert G. Wilson v_, Jun 23 2001