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A224339 Absolute difference between sum of odd divisors of n^2 and sum of even divisors of n^2. 3
1, 5, 13, 29, 31, 65, 57, 125, 121, 155, 133, 377, 183, 285, 403, 509, 307, 605, 381, 899, 741, 665, 553, 1625, 781, 915, 1093, 1653, 871, 2015, 993, 2045, 1729, 1535, 1767, 3509, 1407, 1905, 2379, 3875, 1723, 3705, 1893, 3857, 3751, 2765, 2257, 6617, 2801, 3905, 3991, 5307 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

Multiplicative because A113184 is.

LINKS

Paul D. Hanna, Table of n, a(n) for n = 1..1000

FORMULA

a(n) = (-1)^n * Sum_{d|n^2} (-1)^d * d.

a(n) = A113184(n^2).

Logarithmic derivative of A224340.

a(n) = sigma(n^2) for odd n; a(n) = 4*sigma(n^2/2) - sigma(n^2) for even n. - Andrew Howroyd, Jul 28 2018

EXAMPLE

L.g.f.: L(x) = x + 5*x^2/2 + 13*x^3/3 + 29*x^4/4 + 31*x^5/5 + 65*x^6/6 + 57*x^7/7 + 125*x^8/8 + 121*x^9/9 + 155*x^10/10 +...

where

exp(L(x)) = 1 + x + 3*x^2 + 7*x^3 + 16*x^4 + 30*x^5 + 64*x^6 + 120*x^7 + 236*x^8 + 434*x^9 + 805*x^10 +...+ A224340(n)*x^n +...

MATHEMATICA

dif[n_]:=Module[{divs=Divisors[n^2], od, ev}, od=Total[Select[divs, OddQ]]; ev=Total[Select[divs, EvenQ]]; Abs[od-ev]]; Array[dif, 60] (* Harvey P. Dale, Jul 16 2015 *)

PROG

(PARI) {a(n)=if(n<1, 0, (-1)^n*sumdiv(n^2, d, (-1)^d*d))}

for(n=1, 64, print1(a(n), ", "))

(PARI) a(n) = if(n%2, sigma(n^2), 4*sigma(n^2/2) - sigma(n^2)) \\ Andrew Howroyd, Jul 28 2018

CROSSREFS

Cf. A113184, A224340.

Sequence in context: A321770 A322926 A178854 * A133204 A207040 A309588

Adjacent sequences:  A224336 A224337 A224338 * A224340 A224341 A224342

KEYWORD

nonn,mult

AUTHOR

Paul D. Hanna, Apr 03 2013

STATUS

approved

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Last modified September 26 17:30 EDT 2021. Contains 347670 sequences. (Running on oeis4.)