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A273133 a(n) = n minus the bottom entry of the difference table of the divisors of n.

%I A273133 #33 Mar 13 2020 16:58:42
%S A273133 0,1,1,3,1,4,1,7,5,10,1,11,1,16,7,15,1,6,1,31,13,28,1,36,9,34,19,31,1,
%T A273133 -20,1,31,25,46,7,47,1,52,31,106,1,-62,1,31,21,64,1,151,13,66,43,31,1,
%U A273133 -34,19,8,49,82,1,727,1,88,71,63,25,-6,1,31,61,148,1,12,1,106,11,31,13,22,1,439,65,118,1,1541
%N A273133 a(n) = n minus the bottom entry of the difference table of the divisors of n.
%C A273133 From _David A. Corneth_, May 20 2016: (Start)
%C A273133 The bottom of the difference table of the divisors of n can be expressed in terms of the divisors of n and use of Pascal's triangle. Suppose a, b, c, d and e are the divisors of n. Then the difference table is as follows (rotated for ease of reading):
%C A273133 a
%C A273133 . . b-a
%C A273133 b . . . . c-2b+a
%C A273133 . . c-b . . . . . d-3c+3b-a
%C A273133 c . . . . d-2c+b . . . . . . e-4d+6c-4b+a
%C A273133 . . d-c . . . . . e-3d+3c-b
%C A273133 d . . . . e-2d+c
%C A273133 . . e-d
%C A273133 e
%C A273133 From here we can see Pascal's triangle occurring. Induction can be used to show that it's the case in general.
%C A273133 (End)
%H A273133 David A. Corneth, <a href="/A273133/b273133.txt">Table of n, a(n) for n = 1..10000</a>
%F A273133 a(n) = n - A187202(n).
%F A273133 a(n) = 1, if n is prime.
%F A273133 a(2^k) = 2^k - 1 = A000225(k), k >= 0.
%e A273133 For n = 18 the divisors of 18 are 1, 2, 3, 6, 9, 18, and the difference triangle of the divisors is:
%e A273133 1 . 2 . 3 . 6 . 9 . 18
%e A273133 . 1 . 1 . 3 . 3 . 9
%e A273133 . . 0 . 2 . 0 . 6
%e A273133 . . . 2 .-2 . 6
%e A273133 . . . .-4 . 8
%e A273133 . . . . . 12
%e A273133 The bottom entry is 12, so a(18) = 18 - 12 = 6.
%t A273133 Array[# - First@ NestWhile[Differences, Divisors@ #, Length@ # > 1 &] &, 84] (* _Michael De Vlieger_, May 20 2016 *)
%o A273133 (Sage)
%o A273133 def A273133(n):
%o A273133     D = divisors(n)
%o A273133     T = matrix(ZZ, len(D))
%o A273133     for (m, d) in enumerate(D):
%o A273133         T[0, m] = d
%o A273133         for k in range(m-1, -1, -1) :
%o A273133             T[m-k, k] = T[m-k-1, k+1] - T[m-k-1, k]
%o A273133     return n - T[len(D)-1, 0]
%o A273133 print([A273133(n) for n in range(1, 85)]) # _Peter Luschny_, May 18 2016
%o A273133 (PARI) a(n) = my(d=divisors(n));n-sum(i=1,#d,binomial(#d-1,i-1)*(-1)^(#d-i)*d[i]) \\ _David A. Corneth_, May 20 2016
%Y A273133 Cf. A000005, A000040, A000225, A007318, A187202, A273102, A273103, A273109.
%K A273133 sign
%O A273133 1,4
%A A273133 _Omar E. Pol_, May 17 2016