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A210256
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Differences of the sum of distinct values of {floor(n/k), k=1,...,n}.
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1
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2, 1, 3, 1, 4, 1, 2, 4, 2, 1, 6, 1, 2, 2, 6, 1, 3, 1, 7, 2, 2, 1, 4, 6, 2, 2, 3, 1, 9, 1, 3, 2, 2, 2, 10, 1, 2, 2, 4, 1, 10, 1, 3, 3, 2, 1, 5, 8, 3, 2, 3, 1, 4, 2, 11, 2, 2, 1, 6, 1, 2, 3, 11, 2, 4, 1, 3, 2, 4, 1, 14, 1, 2, 3, 3, 2, 4, 1, 5, 11, 2, 1, 6, 2, 2
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OFFSET
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1,1
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COMMENTS
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It appears that a(n)=1 if and only if n>1 and n+1 is a prime. For example, the indices where 1 occurs in {a(n)} are {2,4,6,10,12,16,...}. Adding 1 to each of these gives {3,5,7,11,13,17,...} each of which is a prime.
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LINKS
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MAPLE
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b:= proc(n) option remember; add(i, i={seq(floor(n/k), k=1..n)}) end:
a:= n-> b(n+1)-b(n):
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MATHEMATICA
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b[n_] := b[n] = Total@ Union@ Table[Floor[n/k], {k, 1, n}];
a[n_] := b[n+1] - b[n];
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PROG
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(Python)
from math import isqrt
def A210256(n): return ((m:=isqrt((n+1<<2)+1)+1>>1)*(m-1)>>1)+sum((n+1)//k for k in range(1, (n+1)//m+1))-((r:=isqrt((n<<2)+1)+1>>1)*(r-1)>>1)-sum(n//k for k in range(1, n//r+1)) # Chai Wah Wu, Oct 31 2023
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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