A228274 a(n) = Sum_{d|n, n/d odd} n * d.

1, 4, 12, 16, 30, 48, 56, 64, 117, 120, 132, 192, 182, 224, 360, 256, 306, 468, 380, 480, 672, 528, 552, 768, 775, 728, 1080, 896, 870, 1440, 992, 1024, 1584, 1224, 1680, 1872, 1406, 1520, 2184, 1920, 1722, 2688, 1892, 2112, 3510, 2208, 2256, 3072, 2793, 3100

Offset: 1

Author

Michael Somos, Aug 19 2013

Keywords

• nonn
• mult

Examples

			G.f. = x + 4*x^2 + 12*x^3 + 16*x^4 + 30*x^5 + 48*x^6 + 56*x^7 + 64*x^8 + ...
a(6) = 48 = 6 * (2 + 6). a(9) = 117 = 9 * (1 + 3 + 9). a(10) = 120 = 10 * (2 + 10).
		

Links

• G. C. Greubel, Table of n, a(n) for n = 1..10000
• M. A. Basoco, On the Fourier developments of a certain class of theta quotients, Bull. Amer. Math. Soc. 49 (1943), 299-306.

Crossrefs

Cf. A002131, A007331, A222171.

Programs

Mathematica

A228274[n_] := If[ n < 1, 0, n Sum[ d Mod[n / d, 2], {d, Divisors @ n}]]; Table[A228274[n], {n, 50}]

PARI

{a(n) = if( n<1, 0, n * sumdiv(n, d, d * (n/d % 2)))};

Formula

Multiplicative with a(2^e) = 4^e, a(p^e) = p^e * (p^(e+1) - 1) / (p - 1) if p>2.

G.f.: Sum_{k>0} k^2 * (x^k + x^(3*k)) / (1 - x^(2*k))^2. [see Basoco (1943) bottom page 305]

G.f.: Sum_{k>0} k^2 * (3 - (-1)^k)/4 * x^k / (1 - x^k)^2.

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

a(n) = n * A002131(n). a(2*n) = 4 * a(n).

a(n) = A007331(n) - 4 * Sum_{k>0} A002131(k) * A002131(n-k). [see Basoco (1943) page 305 equation (9)]

Sum_{k=1..n} a(k) ~ c * n^3, where c = Pi^2/24 = 0.411233... (A222171). - Amiram Eldar, Nov 30 2022