A244048 Antisigma(n) minus the sum of remainders of n mod k, for k = 1,2,3,...,n.
0, 0, 1, 2, 5, 6, 12, 13, 20, 24, 32, 33, 49, 50, 60, 69, 84, 85, 106, 107, 129, 140, 154, 155, 191, 197, 213, 226, 254, 255, 297, 298, 329, 344, 364, 377, 432, 433, 455, 472, 522, 523, 577, 578, 618, 651, 677, 678, 754, 762, 805, 826
Offset: 1
Keywords
Examples
From _Omar E. Pol_, Mar 27 2021: (Start) The following diagrams show a square dissection into regions that are the symmetric representation of A000203, A004125, A153485 and this sequence. In order to construct every diagram we use the following rules: At stage 1 in the first quadrant of the square grid we draw the symmetric representation of sigma(n) using the two Dyck paths described in the rows n and n-1 of A237593. At stage 2 we draw a pair of orthogonal line segments (if it's necessary) such that in the drawing appears totally formed a square n X n. The area of the region that is above the symmetric representation of sigma(n) equals A004125(n). At stage 3 we draw a zig-zag path with line segments of length 1 from (0,n-1) to (n-1,0) such that appears a staircase with n-1 steps. The area of the region (or regions) that is below the symmetric representation of sigma(n) and above the staircase equals a(n). At stage 4 we draw a copy of the symmetric representation of A004125(n) rotated 180 degrees such that one of its vertices is the point (0,0). The area of the region (or regions) that is above of this region and below the staircase equals A153485(n). Illustration for n = 1..6: . _ _ _ _ _ _ . _ _ _ _ _ |_ _ _ |_ R| . _ _ _ _ R |_ _S_| R| | |_T | S |_| . _ _ _ R |_ _ |_| | |_ |_ _| | |_|_ _ | . _ _ |_S_|_| | |_|_S | |_U_|_T | | |_ U |_T | | . _ S |_ S| U|_|_|S| |_ U|_| | | | |_|S| | |_ |_| | . |_| |_|_| |_|_|_| |_|_ _|_| |_V_|_U_|_| |_V_|_ _ _|_| . U V U V . n: 1 2 3 4 5 6 R: A004125 0 0 1 1 4 3 S: A000203 1 3 4 7 6 12 T: a(n) 0 0 1 2 5 6 U: A153485 0 1 2 5 6 12 V: A004125 0 0 1 1 4 3 . Illustration for n = 7..9: . _ _ _ _ _ _ _ _ _ . _ _ _ _ _ _ _ _ |_ _ _S_ _| | . _ _ _ _ _ _ _ |_ _ _ _ | | | |_ |_ _ R | . |_ _S_ _| | | |_ | |_ R | | |_ |_ S| | . | |_ |_ R | | |_ |_S |_ _| | |_ T |_|_ _| . | |_ T |_ _| | |_T |_ _ | |_ _ |_ | | . |_ _ |_ | | |_ _ U |_ | | | | U |_ | | . | |_U |_ |S| | |_ |_ | | | |_ _ |_ |S| . | V | |_| | | V | |_| | | V | |_| | . |_ _ _|_ _ _|_| |_ _ _|_ _ _ _|_| |_ _ _ _|_ _ _ _|_| . n: 7 8 9 R: A004125 8 8 12 S: A000203 8 15 12 T: a(n) 12 13 20 U: A153485 13 20 24 V: A004125 8 8 12 . Illustration for n = 10..12: . _ _ _ _ _ _ _ _ _ _ _ _ . _ _ _ _ _ _ _ _ _ _ _ |_ _ _ _ _ _ | | . _ _ _ _ _ _ _ _ _ _ |_ _ _S_ _ _| | | |_ | |_ _ R | . |_ _ _S_ _ | | | |_ | R | | |_ | |_ | . | |_ | |_ R | | |_ |_ | | |_ |_ S | | . | |_ |_ _|_ | | |_ |_ | | |_ |_ |_ _| . | |_ | |_ _| | |_ T |_ _ _| | |_ T |_ _ _ | . | |_ T |_ _ | |_ _ _ |_ | | |_ _ |_ | | . |_ _ |_ | | | |_ U |_ | | | | U |_ | | . | |_ U |_ |S| | |_ |_ |S| | |_ |_ | | . | |_ |_ | | | | |_ | | | |_ _ |_ | | . | V | |_| | | V | |_| | | V | |_| | . |_ _ _ _|_ _ _ _ _|_| |_ _ _ _ _|_ _ _ _ _|_| |_ _ _ _ _|_ _ _ _ _ _|_| . n: 10 11 12 R: A004125 13 22 17 S: A000203 18 12 28 T: a(n) 24 32 33 U: A153485 32 33 49 V: A004125 13 22 17 . Note that in the diagrams the symmetric representation of a(n) is the same as the symmetric representation of A153485(n-1) rotated 180 degrees. The original examples (dated Jun 24 2014) were only the diagrams for n = 11 and n = 12. (End)
Links
- Paolo Xausa, Table of n, a(n) for n = 1..10000
Crossrefs
Programs
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Mathematica
With[{r=Range[100]},Join[{0},Accumulate[DivisorSigma[1,r]-r]]] (* Paolo Xausa, Oct 16 2023 *) -
Python
from math import isqrt def A244048(n): return (-n*(n-1)-(s:=isqrt(n-1))**2*(s+1) + sum((q:=(n-1)//k)*((k<<1)+q+1) for k in range(1,s+1)))>>1 # Chai Wah Wu, Oct 22 2023
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