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A135768 Indices of pentagonal numbers > 0 which are not the difference of 2 other pentagonal numbers > 0.

%I A135768 #22 Feb 05 2016 20:37:37
%S A135768 1,2,3,6,8,9,11,15,18,24,27,54,81,96,128,135,162,216,243,288,303,384,
%T A135768 423,459,486,519,591,639,648,683,729,783,864,879,891,1215,1458,1719,
%U A135768 1944,2031,2043,2048,2151,2187,2463,2799,3231,3456,3543,3879,3903,4023
%N A135768 Indices of pentagonal numbers > 0 which are not the difference of 2 other pentagonal numbers > 0.
%C A135768 A subsequence of A136112, obtained by omitting A136112(A135771(k)), k=1,2,3,... ; i.e. those which are not the difference of two larger pentagonal numbers, but the difference of a larger and a smaller (or equal) pentagonal number. Sequence A135769 has the pentagonal numbers corresponding to these indices.
%H A135768 Donovan Johnson and Chai Wah Wu, <a href="/A135768/b135768.txt">Table of n, a(n) for n = 1..10000</a> n = 1..200 from Donovan Johnson
%F A135768 P(n)=n*(3*n-1)/2 <=> n*(n-1/3) = (2/3)*P(n), thus m = P(n) <=> m = P([sqrt(2m/3)]+1) and m = P(n) <=> 24m+1 = (6n-1)^2, useful for investigating the possibility of write P(n)=P(n')+P(n"): this is possible whenever (6n-1)^2 = (6n'-1)^2 + (6n"-1)^2.
%e A135768 Indices of the following numbers are not here but in A136112:
%e A135768 P_5 = P_7 - P_5
%e A135768 P_23 = P_24 - P_7
%e A135768 P_51 = P_66 - P_42
%e A135768 P_71 = P_74 - P_21
%e A135768 P_72 = P_80 - P_35
%e A135768 P_99 = P_104 - P_32
%e A135768 P_123 = P_144 - P_75
%e A135768 P_239 = P_249 - P_70
%e A135768 P_263 = P_274 - P_77
%e A135768 P_311 = P_324 - P_91
%e A135768 P_359 = P_374 - P_105
%t A135768 Select[Range[100], Reduce[# (3 # - 1) == x (3 x - 1) - y (3 y - 1) && x > 0 && y > 0, {x, y}, Integers] == False &] (* _T. D. Noe_, Dec 05 2011 *)
%o A135768 (PARI) P(n)=n*(3*n-1)/2
%o A135768 isPent(t)=P(sqrtint((t*2)\3)+1)==t
%o A135768 for( i=1,999,for( j=1,(P(i)-1)\3, isPent(P(i)+P(j))&next(2)); print1(i","))
%Y A135768 Cf. A000326, A136112-A136118, A135769(n) = A000326(a(n)), A135771 = A136112 \ A135768.
%K A135768 nonn
%O A135768 1,2
%A A135768 _R. J. Mathar_ and _M. F. Hasler_, Feb 07 2008
%E A135768 Extended by T. D. Noe, Dec 05 2011