A000385 Convolution of A000203 with itself. (Formerly M4113 N1708)

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

Offset

1,2

Comments

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

References

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.

Links

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]

Formula

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

Maple

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

Mathematica

a[n_] := Sum[DivisorSigma[1, k] DivisorSigma[1, n-k+1], {k, 1, n}];
Array[a, 100] (* Jean-François Alcover, Aug 01 2018 *)

Prog

(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

Crossrefs

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

Keyword

nonn,easy,look

Author

N. J. A. Sloane

Extensions

More terms from Sean A. Irvine, Nov 14 2010

Status

approved