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A325459 Sum of numbers of nontrivial divisors (greater than 1 and less than k) of k for k = 1..n.

%I A325459 #37 Oct 08 2021 03:44:35
%S A325459 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,
%T A325459 42,46,46,52,52,56,58,60,62,69,69,71,73,79,79,85,85,89,93,95,95,103,
%U A325459 104,108,110,114,114,120,122,128,130,132,132,142
%N A325459 Sum of numbers of nontrivial divisors (greater than 1 and less than k) of k for k = 1..n.
%C A325459 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).
%C A325459 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.
%H A325459 Alois P. Heinz, <a href="/A325459/b325459.txt">Table of n, a(n) for n = 0..10000</a>
%F A325459 From _M. F. Hasler_, Oct 11 2019: (Start)
%F A325459 a(n) = A006218(n) - 2*n + 1, in terms of partial sums of number of divisors.
%F A325459 a(n) = Sum_{k=1..n} A070824(k): partial sums of A070824 = number of nontrivial divisors. (End)
%e A325459 The a(4) = 1 through a(10) = 8 partitions:
%e A325459   (22)  (221)  (33)    (331)    (44)      (333)      (55)
%e A325459                (222)   (2221)   (2222)    (441)      (3331)
%e A325459                (2211)  (22111)  (3311)    (22221)    (4411)
%e A325459                                 (22211)   (33111)    (22222)
%e A325459                                 (221111)  (222111)   (222211)
%e A325459                                           (2211111)  (331111)
%e A325459                                                      (2221111)
%e A325459                                                      (22111111)
%p A325459 a:= proc(n) option remember; `if`(n<2, 0,
%p A325459       numtheory[tau](n)-2+a(n-1))
%p A325459     end:
%p A325459 seq(a(n), n=0..100);  # _Alois P. Heinz_, Oct 11 2019
%t A325459 Table[Length[Select[IntegerPartitions[n],MatchQ[#,{x_,y__,1...}/;x>1&&SameQ[x,y]]&]],{n,0,30}]
%t A325459 (* Second program: *)
%t A325459 a[n_] := a[n] = If[n<2, 0, DivisorSigma[0, n] - 2 + a[n-1]];
%t A325459 a /@ Range[0, 100] (* _Jean-François Alcover_, May 20 2021, after _Alois P. Heinz_ *)
%o A325459 (Python)
%o A325459 from math import isqrt
%o A325459 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
%Y A325459 Cf. A049988, A093641, A325349, A325351, A325355, A325356, A325357, A325358.
%Y A325459 Cf. A070824, A006218.
%K A325459 nonn
%O A325459 0,7
%A A325459 _Gus Wiseman_, May 04 2019
%E A325459 Name changed at the suggestion of _Patrick James Smalley-Wall_ and _Luc Rousseau_ by _Gus Wiseman_, Oct 11 2019