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A000385 Convolution of A000203 with itself.

%I A000385 M4113 N1708 #71 Jan 04 2025 06:19:53
%S A000385 1,6,17,38,70,116,185,258,384,490,686,826,1124,1292,1705,1896,2491,
%T A000385 2670,3416,3680,4602,4796,6110,6178,7700,7980,9684,9730,12156,11920,
%U A000385 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 A000385 Convolution of A000203 with itself.
%C A000385 Convolution of A340793 and A024916. - _Omar E. Pol_, Feb 17 2021
%D A000385 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D A000385 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%D A000385 Jacques Touchard, On prime numbers and perfect numbers, Scripta Math., 129 (1953), 35-39.
%H A000385 Reinhard Zumkeller, <a href="/A000385/b000385.txt">Table of n, a(n) for n = 1..10000</a>
%H A000385 Nicolae Ciprian Bonciocat, <a href="http://www.hms.gr/apothema/?s=sa&amp;i=762">Congruences for the Convolution of Divisor sum function</a>, Bull. Greek Math. Soc., Vol 47 (2003), pp. 19-29.
%H A000385 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 A000385 Srinivasa 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 A000385 Jacques Touchard, <a href="/A000385/a000385.pdf">On prime numbers and perfect numbers</a>, Scripta Math., 129 (1953), 35-39. [Annotated scanned copy]
%F A000385 a(n) = Sum_{k=1..n} A000203(k)*A000203(n-k+1).
%F A000385 G.f.: (1/x)*(Sum_{k>=1} k*x^k/(1 - x^k))^2. - _Ilya Gutkovskiy_, Nov 10 2016
%F A000385 a(5*n+1)==0 (mod 5) and a(7*n+6)==0 (mod 7). See Bonciocat link. - _Michel Marcus_, Nov 10 2016
%F A000385 a(n) = (5/12)*A001158(n+1) - ((5+6*n)/12)*A000203(n+1). - _Robert Israel_, Sep 17 2018
%F A000385 Sum_{k=1..n} a(k) ~ Pi^4 * n^4 / 864. - _Vaclav Kotesovec_, Apr 02 2019
%p A000385 f:= n -> 5/12*numtheory:-sigma[3](n+1)-(5+6*n)/12*numtheory:-sigma(n+1):
%p A000385 map(f, [$1..100]); # _Robert Israel_, Sep 17 2018
%t A000385 a[n_] := Sum[DivisorSigma[1, k] DivisorSigma[1, n-k+1], {k, 1, n}];
%t A000385 Array[a, 100] (* _Jean-François Alcover_, Aug 01 2018 *)
%o A000385 (Haskell)
%o A000385 a000385 n = sum $ zipWith (*) sigmas $ reverse sigmas where
%o A000385    sigmas = take n a000203_list
%o A000385 -- _Reinhard Zumkeller_, Sep 20 2011
%o A000385 (PARI) a(n) = sum(k=1, n, sigma(k)*sigma(n-k+1)); \\ _Michel Marcus_, Nov 10 2016
%o A000385 (PARI) a(n) = my(f = factor(n+1)); (5 * sigma(f, 3) - (6*n + 5) * sigma(f)) / 12; \\ _Amiram Eldar_, Jan 04 2025
%o A000385 (Python)
%o A000385 from sympy import factorint
%o A000385 def A000385(n):
%o A000385     f = factorint(n+1).items()
%o A000385     return(5*prod((p**(3*(e+1))-1)//(p**3-1) for p,e in f)-(5+6*n)*prod((p**(e+1)-1)//(p-1) for p, e in f))//12 # _Chai Wah Wu_, Jul 25 2024
%Y A000385 Cf. A000203, A024916, A001158, A340793.
%Y A000385 Column k=2 of A319083 (shifted).
%K A000385 nonn,easy,look
%O A000385 1,2
%A A000385 _N. J. A. Sloane_
%E A000385 More terms from _Sean A. Irvine_, Nov 14 2010