A000385 Convolution of A000203 with itself.

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

Author

N. J. A. Sloane

Keywords

• nonn
• easy
• look

Comments

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).
• Jacques 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., Vol 47 (2003), pp. 19-29.
• MathOverflow, Searching for a proof for a series identity.
• Srinivasa Ramanujan, On certain arithmetical functions, Transactions of the Cambridge Philosophical Society, 22, No.9 (1916), 169- 184 (see Table IV, line 1).
• Jacques Touchard, On prime numbers and perfect numbers, Scripta Math., 129 (1953), 35-39. [Annotated scanned copy]

Crossrefs

Cf. A000203, A024916, A001158, A340793.

Column k=2 of A319083 (shifted).

Programs

Haskell

a000385 n = sum $ zipWith (*) sigmas $ reverse sigmas where
   sigmas = take n a000203_list
-- Reinhard Zumkeller, Sep 20 2011

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 *)

PARI

a(n) = sum(k=1, n, sigma(k)*sigma(n-k+1)); \\ Michel Marcus, Nov 10 2016

PARI

a(n) = my(f = factor(n+1)); (5 * sigma(f, 3) - (6*n + 5) * sigma(f)) / 12; \\ Amiram Eldar, Jan 04 2025

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

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(5*n+1)==0 (mod 5) and a(7*n+6)==0 (mod 7). See Bonciocat link. - Michel Marcus, 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

Extensions

More terms from Sean A. Irvine, Nov 14 2010