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A000385
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Convolution of A000203 with itself.
(Formerly M4113 N1708)
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25
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1, 6, 17, 38, 70, 116, 185, 258, 384, 490, 686, 826, 1124, 1292, 1705, 1896, 2491, 2670, 3416, 3680, 4602, 4796, 6110, 6178, 7700, 7980, 9684, 9730, 12156, 11920, 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
(list;
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refs;
listen;
history;
text;
internal format)
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OFFSET
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1,2
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COMMENTS
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a(5*n+1)==0 (mod 5) and a(7*n+6)==0 (mod 7). See Bonciocat link. - Michel Marcus, Nov 10 2016
Convolution of A340793 and A024916. - Omar E. Pol, Feb 17 2021
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REFERENCES
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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).
J. Touchard, On prime numbers and perfect numbers, Scripta Math., 129 (1953), 35-39.
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LINKS
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Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Nicolae Ciprian Bonciocat, Congruences for the Convolution of Divisor sum function, Bull. Greek Math. Soc., p. 19-29, Vol 47, 2003.
MathOverflow, Searching for a proof for a series identity
S. Ramanujan, On certain arithmetical functions, Transactions of the Cambridge Philosophical Society, 22, No.9 (1916), 169- 184 (see Table IV, line 1).
J. Touchard, On prime numbers and perfect numbers, Scripta Math., 129 (1953), 35-39. [Annotated scanned copy]
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FORMULA
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a(n) = Sum_{k=1..n} A000203(k)*A000203(n-k+1).
G.f.: (1/x)*(Sum_{k>=1} k*x^k/(1 - x^k))^2. - Ilya Gutkovskiy, Nov 10 2016
a(n) = (5/12)*A001158(n+1) - ((5+6*n)/12)*A000203(n+1). - Robert Israel, Sep 17 2018
Sum_{k=1..n} a(k) ~ Pi^4 * n^4 / 864. - Vaclav Kotesovec, Apr 02 2019
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MAPLE
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f:= n -> 5/12*numtheory:-sigma[3](n+1)-(5+6*n)/12*numtheory:-sigma(n+1):
map(f, [$1..100]); # Robert Israel, Sep 17 2018
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MATHEMATICA
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a[n_] := Sum[DivisorSigma[1, k] DivisorSigma[1, n-k+1], {k, 1, n}];
Array[a, 100] (* Jean-François Alcover, Aug 01 2018 *)
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PROG
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(Haskell)
a000385 n = sum $ zipWith (*) sigmas $ reverse sigmas where
sigmas = take n a000203_list
-- Reinhard Zumkeller, Sep 20 2011
(PARI) a(n) = sum(k=1, n, sigma(k)*sigma(n-k+1)); \\ Michel Marcus, Nov 10 2016
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CROSSREFS
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Cf. A000203, A024916, A001158, A340793.
Column k=2 of A319083 (shifted).
Sequence in context: A212980 A132127 A023621 * A192756 A004799 A085278
Adjacent sequences: A000382 A000383 A000384 * A000386 A000387 A000388
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KEYWORD
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nonn,easy,look
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AUTHOR
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N. J. A. Sloane
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EXTENSIONS
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More terms from Sean A. Irvine, Nov 14 2010
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STATUS
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approved
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