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

0, 0, 2, 3, 6, 9, 16, 13, 27, 31, 34, 43, 57, 56, 75, 80, 96, 99, 121, 122, 155, 164, 163, 184, 220, 218, 255, 252, 277, 304, 339, 328, 372, 389, 412, 433, 491, 478, 515, 536, 570, 609, 638, 647, 722, 713, 746, 767, 858, 842, 910, 939, 942, 993, 1060, 1057

Offset: 1

Author

Amarnath Murthy, Jan 29 2002

Keywords

• easy
• nonn

Examples

			a(7) = 16 = 1 +3 +6 +4 +2 = 13 % 3 + 13 % 5 + 13 % 7 + 13 % 9 + 13 % 11.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000203, A000593, A004125, A067436, A024916, A327329.

Programs

Maple

L:= [seq(4*n-3 - numtheory:-sigma(2*n-1)-numtheory:-sigma((n-1)/2^padic:-ordp(n-1,2)), n=1..100)]:
ListTools:-PartialSums(L); # Robert Israel, Jan 16 2019

Python

from math import isqrt
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))
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

Formula

a(n) = a(n-1) + 4*n-3 - A000203(2*n-1) - A000593(n-1). - Robert Israel, Jan 16 2019

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

Extensions

Corrected and extended by several contributors.