A189835 Number of representations of n as a*b + b*c + c*d + d*e where a, b, d, e>0, c>=0 are integers.

0, 1, 4, 9, 16, 26, 36, 53, 64, 90, 100, 138, 144, 194, 200, 261, 256, 347, 324, 426, 416, 522, 484, 658, 576, 746, 712, 882, 784, 1060, 900, 1173, 1088, 1314, 1160, 1587, 1296, 1658, 1544, 1890, 1600, 2164, 1764, 2298, 2096, 2466, 2116, 2930, 2304, 2955, 2696

Offset

1,3

Comments

Related to "Liouville's Last Theorem".

Links

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

R. P. Agarwal, Lambert series and Ramanujan, Prod. Indian Acad. Sci. (Math. Sci.), v. 103, n. 3, 1993, pp. 269-293. see p. 273.

George E. Andrews, Stacked lattice boxes, Ann. Comb. 3 (1999), 115-130. See L_5(n).

E. T. Bell, The form wx+xy+yz+zu, Bull. Amer. Math. Soc., 42 (1936), 377-380.

Formula

G.f.: Sum_{k>0} (x^k + x^(2*k)) / (1 - x^k)^3 - k * x^k / (1 - x^k)^2.

a(n) = A001157(n) - A038040(n) = sigma( n, 2) - n * sigma( n, 0) where sigma( n, k) is the sum of the k-th powers of the divisors of n.

Example

G.f. = x^2 + 4*x^3 + 9*x^4 + 16*x^5 + 26*x^6 + 36*x^7 + 53*x^8 + 64*x^9 + 90*x^10 + ...
a(3) = 4 since 3 = 1*1 + 1*0 + 0*1 + 1*2 = 1*1 + 1*0 + 0*2 + 2*1 = 1*2 + 2*0 + 0*1 + 1*1 = 2*1 + 1*0 + 0*1 + 1*1 are all 4 representations of 3.

Maple

with(numtheory); f:=n->sigma[2](n)-n*sigma[0](n);

Mathematica

a[n_] := DivisorSigma[2, n] - n*DivisorSigma[0, n]; Table[a[n], {n, 51}] (* Jean-François Alcover, Aug 31 2011 *)

Prog

(PARI) {a(n) = if( n<1, 0, sigma( n, 2) - n * sigma( n, 0))}
(Haskell)
a189835 n = a001157 n - a038040 n  -- Reinhard Zumkeller, Jan 21 2014

Crossrefs

Cf. A001157, A038040, A191822.

Sequence in context: A274963 A353387 A050461 * A183764 A022334 A073171

Adjacent sequences: A189832 A189833 A189834 * A189836 A189837 A189838

Keyword

nonn

Author

Michael Somos, Apr 28 2011

Extensions

Added references, comment, Maple program, cross-reference to A191822. - N. J. A. Sloane, Jun 17 2011

Status

approved