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%I M4113 N1708 #57 Feb 17 2021 14:30:14
%S 1,6,17,38,70,116,185,258,384,490,686,826,1124,1292,1705,1896,2491,
%T 2670,3416,3680,4602,4796,6110,6178,7700,7980,9684,9730,12156,11920,
%U 14601,14752,17514,17224,21395,20406,24590,24556,28920,27860,34112,32186,38674,37994,43980,42136,51646,47772,56749,55500,64316,60606,73420,67956,80500,77760,88860,83810,102284,92690,108752,105236,120777,112672,135120,123046,145194,138656,157512,146580,177515,159396,185744,179122
%N Convolution of A000203 with itself.
%C a(5*n+1)==0 (mod 5) and a(7*n+6)==0 (mod 7). See Bonciocat link. - _Michel Marcus_, Nov 10 2016
%C Convolution of A340793 and A024916. - _Omar E. Pol_, Feb 17 2021
%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).
%D J. Touchard, On prime numbers and perfect numbers, Scripta Math., 129 (1953), 35-39.
%H Reinhard Zumkeller, <a href="/A000385/b000385.txt">Table of n, a(n) for n = 1..10000</a>
%H Nicolae Ciprian Bonciocat, <a href="http://www.hms.gr/apothema/?s=sa&i=762">Congruences for the Convolution of Divisor sum function</a>, Bull. Greek Math. Soc., p. 19-29, Vol 47, 2003.
%H MathOverflow, <a href="https://mathoverflow.net/questions/310735/searching-for-a-proof-for-a-series-identity">Searching for a proof for a series identity</a>
%H S. Ramanujan, <a href="http://ramanujan.sirinudi.org/Volumes/published/ram18.pdf">On certain arithmetical functions</a>, Transactions of the Cambridge Philosophical Society, 22, No.9 (1916), 169- 184 (see Table IV, line 1).
%H J. Touchard, <a href="/A000385/a000385.pdf">On prime numbers and perfect numbers</a>, Scripta Math., 129 (1953), 35-39. [Annotated scanned copy]
%F a(n) = Sum_{k=1..n} A000203(k)*A000203(n-k+1).
%F G.f.: (1/x)*(Sum_{k>=1} k*x^k/(1 - x^k))^2. - _Ilya Gutkovskiy_, Nov 10 2016
%F a(n) = (5/12)*A001158(n+1) - ((5+6*n)/12)*A000203(n+1). - _Robert Israel_, Sep 17 2018
%F Sum_{k=1..n} a(k) ~ Pi^4 * n^4 / 864. - _Vaclav Kotesovec_, Apr 02 2019
%p f:= n -> 5/12*numtheory:-sigma[3](n+1)-(5+6*n)/12*numtheory:-sigma(n+1):
%p map(f, [$1..100]); # _Robert Israel_, Sep 17 2018
%t a[n_] := Sum[DivisorSigma[1, k] DivisorSigma[1, n-k+1], {k, 1, n}];
%t Array[a, 100] (* _Jean-François Alcover_, Aug 01 2018 *)
%o (Haskell)
%o a000385 n = sum $ zipWith (*) sigmas $ reverse sigmas where
%o sigmas = take n a000203_list
%o -- _Reinhard Zumkeller_, Sep 20 2011
%o (PARI) a(n) = sum(k=1, n, sigma(k)*sigma(n-k+1)); \\ _Michel Marcus_, Nov 10 2016
%Y Cf. A000203, A024916, A001158, A340793.
%Y Column k=2 of A319083 (shifted).
%K nonn,easy,look
%O 1,2
%A _N. J. A. Sloane_
%E More terms from _Sean A. Irvine_, Nov 14 2010
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