A330856 - OEIS

%I #11 Apr 29 2020 14:30:16

%S 0,2,4,11,15,25,33,48,56,75,87,111,127,149,165,204,220,251,277,315,

%T 339,383,407,459,491,536,564,628,660,714,762,825,857,923,959,1046,

%U 1098,1156,1196,1294,1342,1416,1480,1560,1608,1710,1758,1866,1930,2018,2080,2194,2250

%N Total sum of divisors of all the parts in the partitions of n into 2 parts.

%H Robert Israel, <a href="/A330856/b330856.txt">Table of n, a(n) for n = 1..10000</a>

%H <a href="/index/Par#part">Index entries for sequences related to partitions</a>

%F a(n) = Sum_{i=1..floor(n/2)} sigma(i) + sigma(n-i), where sigma(n) is the sum of divisors of n (A000203).

%F a(n) = ((n+1) mod 2) * sigma(floor(n/2)) + Sum_{i=1..n-1} sigma(i), where sigma(n) is the sum of divisors of n (A000203).

%e a(4) = 11; 4 has two partitions into 2 parts, (3,1) and (2,2). The total sum of all divisors of the parts is sigma(3) + sigma(1) + sigma(2) + sigma(2) = 4 + 1 + 3 + 3 = 11.

%p N:= 100: # for a(1) ... a(N)

%p S:= map(numtheory:-sigma, [$1..N]):

%p T:= ListTools:-PartialSums(S):

%p [0, seq(T[i-1]+`if`(i::even, S[i/2],0),i=2..N)]; # _Robert Israel_, Apr 29 2020

%t Table[Sum[DivisorSigma[1, i] + DivisorSigma[1, n - i], {i, Floor[n/2]}], {n, 80}]

%Y Cf. A000203, A330857 (distinct parts).

%K nonn,easy

%O 1,2

%A _Wesley Ivan Hurt_, Apr 27 2020