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
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
G.f.: (1/x)*(Sum_{k>=1} k*x^k/(1 - x^k))^2. - Ilya Gutkovskiy, Nov 10 2016
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
(Python)
from sympy import factorint
def A000385(n):
f = factorint(n+1).items()
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
CROSSREFS
KEYWORD
AUTHOR
EXTENSIONS
More terms from Sean A. Irvine, Nov 14 2010
STATUS
approved