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A067435 a(n) is the sum of all the remainders when n-th odd number is divided by odd numbers < 2n-1.

%I A067435 #18 Nov 01 2023 18:30:23
%S A067435 0,0,2,3,6,9,16,13,27,31,34,43,57,56,75,80,96,99,121,122,155,164,163,
%T A067435 184,220,218,255,252,277,304,339,328,372,389,412,433,491,478,515,536,
%U A067435 570,609,638,647,722,713,746,767,858,842,910,939,942,993,1060,1057
%N A067435 a(n) is the sum of all the remainders when n-th odd number is divided by odd numbers < 2n-1.
%H A067435 Robert Israel, <a href="/A067435/b067435.txt">Table of n, a(n) for n = 1..10000</a>
%F A067435 a(n)  = a(n-1) + 4*n-3 - A000203(2*n-1) - A000593(n-1). - _Robert Israel_, Jan 16 2019
%F A067435 a(n) = n*(2*n-1) - A326123(n) - A078471(n-1) = n*(2*n-1) - A024916(2*n) - 2*A024916(floor(n/2)) + 3*A024916(n) + 2*A024916(floor((n-1)/2)) - A024916(n-1). - _Chai Wah Wu_, Nov 01 2023
%e A067435 a(7) = 16 = 1 +3 +6 +4 +2 = 13 % 3 + 13 % 5 + 13 % 7 + 13 % 9 + 13 % 11.
%p A067435 L:= [seq(4*n-3 - numtheory:-sigma(2*n-1)-numtheory:-sigma((n-1)/2^padic:-ordp(n-1,2)), n=1..100)]:
%p A067435 ListTools:-PartialSums(L); # _Robert Israel_, Jan 16 2019
%o A067435 (Python)
%o A067435 from math import isqrt
%o A067435 def A327329(n): return -(s:=isqrt(n))**2*(s+1)+sum((q:=n//k)*((k<<1)+q+1) for k in range(1,s+1))
%o A067435 def A067435(n): return n*((n<<1)-1)-(A327329(n<<1)>>1)-A327329(n>>1)+3*(A327329(n)>>1)+A327329(n-1>>1)-(A327329(n-1)>>1) # _Chai Wah Wu_, Nov 01 2023
%Y A067435 Cf. A000203, A000593, A004125, A067436, A024916, A327329.
%K A067435 easy,nonn
%O A067435 1,3
%A A067435 _Amarnath Murthy_, Jan 29 2002
%E A067435 Corrected and extended by several contributors.