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A325459
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Sum of numbers of nontrivial divisors (greater than 1 and less than k) of k for k = 1..n.
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4
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0, 0, 0, 0, 1, 1, 3, 3, 5, 6, 8, 8, 12, 12, 14, 16, 19, 19, 23, 23, 27, 29, 31, 31, 37, 38, 40, 42, 46, 46, 52, 52, 56, 58, 60, 62, 69, 69, 71, 73, 79, 79, 85, 85, 89, 93, 95, 95, 103, 104, 108, 110, 114, 114, 120, 122, 128, 130, 132, 132, 142
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internal format)
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OFFSET
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0,7
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COMMENTS
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Also the number of integer partitions of n that are not hooks but whose augmented differences are hooks (original name). The augmented differences aug(y) of an integer partition y of length k are given by aug(y)_i = y_i - y_{i + 1} + 1 if i < k and otherwise aug(y)_k = y_k. For example, aug(6,5,5,3,3,3) = (2,1,3,1,1,3).
This sequence counts integer partitions with any number of ones and one part > 1 which appears at least twice. The Heinz numbers of these partitions are given by A325359.
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LINKS
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FORMULA
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a(n) = A006218(n) - 2*n + 1, in terms of partial sums of number of divisors.
a(n) = Sum_{k=1..n} A070824(k): partial sums of A070824 = number of nontrivial divisors. (End)
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EXAMPLE
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The a(4) = 1 through a(10) = 8 partitions:
(22) (221) (33) (331) (44) (333) (55)
(222) (2221) (2222) (441) (3331)
(2211) (22111) (3311) (22221) (4411)
(22211) (33111) (22222)
(221111) (222111) (222211)
(2211111) (331111)
(2221111)
(22111111)
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MAPLE
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a:= proc(n) option remember; `if`(n<2, 0,
numtheory[tau](n)-2+a(n-1))
end:
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MATHEMATICA
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Table[Length[Select[IntegerPartitions[n], MatchQ[#, {x_, y__, 1...}/; x>1&&SameQ[x, y]]&]], {n, 0, 30}]
(* Second program: *)
a[n_] := a[n] = If[n<2, 0, DivisorSigma[0, n] - 2 + a[n-1]];
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PROG
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(Python)
from math import isqrt
def A325459(n): return 0 if n == 0 else (lambda m: 2*(sum(n//k for k in range(1, m+1))-n)+(1-m)*(1+m))(isqrt(n)) # Chai Wah Wu, Oct 07 2021
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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