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%I #69 Jul 02 2023 16:32:30
%S 2,4,6,12,20,30,42,56,72,90,110,132,156,182,210,240,272,306,342,380,
%T 420,462,506,552,600,650,702,756,812,870,930,992,1056,1122,1190,1260,
%U 1332,1406,1482,1560,1640,1722,1806,1892,1980,2070,2162,2256,2352,2450
%N Numbers k such that difference between the upper and lower central divisors of k is 1.
%C From _David A. Corneth_, Sep 02 2018: (Start)
%C Theorem: a(1) = 2, a(2) = 4; a(n) = n*(n-1) for n > 2.
%C Proof:
%C If a(n) is a square m^2 then the upper central divisor is m and by definition of the sequence the lower one is m-1. But m-1 and m are coprime, and (m-1)|m^2 implies m-1 = 1, i.e. a(n) = 4.
%C if a(n) is not a square then it has an even number of divisors with m and m-1 the central divisors, so it has the form m*(m-1), i.e. is oblong (see A002378). QED (End)
%H Harvey P. Dale, <a href="/A060798/b060798.txt">Table of n, a(n) for n = 1..2000</a> [First 1000 terms from Harry J. Smith]
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3, -3, 1).
%F Solutions to A033677(k) - A060775(k) = 1, where k = j*(j+1) and at least one of j and j+1 is composite.
%F Except at n < 5, this sequence satisfies a(n+1) = 3*a(n) - 3*a(n-1) + a(n-2). - _Georgi Guninski_, Jun 07 2010 [This follows from Corneth's theorem above. - _N. J. A. Sloane_, Sep 02 2018]
%F G.f.: (2*x^2-2*x+1)*(x^3-x^2-x-1) / (x-1)^3. - _Colin Barker_, Apr 16 2014 [This follows from Corneth's theorem above. - _N. J. A. Sloane_, Sep 02 2018]
%e The divisors of 2 are 1 and 2, so the upper central divisor is 2 and the lower central divisor is 1, so a(1)=2 is a member.
%e k = 4032 = 2*2*2*2*2*2*3*3*7 is here because its central divisors (the 21st and 22nd divisors) are {63,64} which differ by 1.
%t dulcdQ[n_]:=Module[{d=Divisors[n],len},len=Floor[Length[d]/2];d[[len+1]] - d[[len]]==1]; Select[Range[2500],dulcdQ] (* or *) Join[{2,4},Table[ n(n-1),{n,3,60}]] (* after David A. Corneth's comment and formula *) (* _Harvey P. Dale_, Aug 28 2018 *)
%o (PARI) { n=-1; for (m=1, 999000, d=divisors(m); if (m==1 || (d[1 + length(d)\2] - d[length(d)\2]) == 1, write("b060798.txt", n++, " ", m)); ) } \\ _Harry J. Smith_, Jul 13 2009
%o (PARI) first(n) = res = List([2, 4]); for(i = 3, n, listput(res, i*(i-1))); res \\ _David A. Corneth_, Sep 02 2018
%Y Cf. A000196, A002378, A033677, A060775.
%K nonn,easy
%O 1,1
%A _Labos Elemer_, Apr 27 2001
%E Start and offset changed by _N. J. A. Sloane_, Sep 02 2018 at the suggestion of _Harvey P. Dale_. Further edited by _N. J. A. Sloane_, Sep 02 2018
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