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%I #10 Jul 14 2015 01:28:53
%S 1,4,9,12,16,24,25,36,40,45,49,60,64,72,81,84,100,105,112,121,144,160,
%T 169,180,189,196,216,220,225,240,256,264,280,289,297,300,312,324,352,
%U 360,361,364,385,396,400,420,429,432,441,480,484,504,520,529,544,576
%N Numbers n with at least one solution to n=x*y*z, x+y-z=1 with ordered triples (x,y,z), x,y,z>=1.
%C Supersequence of A000290, i.e., all perfect squares are in the sequence.
%C Solutions (x,y) are integral points on the elliptic curve x*y*(x+y-1)=n. - _Georgi Guninski_, Oct 25 2010
%C From _Robert G. Wilson v_, Oct 25 2010: (Start)
%C a(n) != 2 (mod 3) nor {2, 3} (mod 4) nor 3 (mod 5). a(n) == {0, 1, 4, 9, 12, 16, 21, 24, 25, 36, 37, 40, 45, 49, 52, 57} (mod 60).
%C Terms which are congruent to {0, 1, 4, 9, 12, 16, 21, 24, 25, 36, 37, 40, 45, 49, 52, 57} (mod 60) and are not members of the sequence: 21, 37, 52, 57, 61, 69, 76, 85, 96, 97, 109, 117, 120, 124, 129, 132, 136, 141, 145, 156, 157, 165, 172, 177, 181, ..., .
%C Terms which are congruent to {0, 1, 4, 9, 12, 16, 21, 24, 25, 36, 37, 40, 45, 49, 52, 57} (mod 60), not prime and are not members of the sequence: 21, 52, 57, 69, 76, 85, 96, 117, 120, 124, 129, 132, 136, 141, 145, 156, 165, 172, 177, 184, 192, 201, ..., .
%C Nonsquare terms: 12, 24, 40, 45, 60, 72, 84, 105, 112, 160, 180, 189, 216, 220, 240, 264, 280, 297, 300, ..., .
%C The lesser of twin terms: 24, 360, 624, 840, 960, 1104, 1224, 2184, 2400, 2736, ..., .
%C Lesser term of a gap of n or 0 if impossible: 24, 0, 1, 12, 4, 0, 105, 16, 72, 0, 25, ..., . (End)
%C Number of terms less than or equal to 10^n: 1, 3, 17, 84, 423, 2123, 10603, 52144, 253257, ..., . - _Robert G. Wilson v_, Oct 30 2010
%H Robert G. Wilson v, <a href="/A171920/b171920.txt"> Table of n, a(n) for n = 1..10603 </a>. [From _Robert G. Wilson v_, Oct 25 2010]
%e n=1 allows a solution (x,y,z)=(1,1,1), and is in the sequence.
%e n=4 allows solutions (x,y,z)=(1,2,2) and (2,1,2) and is in the sequence.
%t fQ[n_] := Block[{c = 0, cong = {0, 1, 4, 9, 12, 16, 21, 24, 25, 36, 37, 40, 45, 49, 52, 57}, dvs = Divisors@ n, dvt, j = 1, k, lmt1, lmt2}, If[ MemberQ[ cong, Mod[n, 60]], lmtj = Length@ dvs + 1; While[j < lmtj, dvt = Divisors[ n/dvs[[j]]]; k = 1; lmtk = Length@ dvt + 1; While[k < lmtk, If[ dvs[[j]] + dvt[[k]] == n/(dvs[[j]]*dvt[[k]]) + 1, c++ ]; k++ ]; j++ ]]; c > 0]; Select[ Range@ 584, fQ] (* _Robert G. Wilson v_, Oct 25 2010 *)
%o (PARI) is_A171920(n)={ my(L=sqrt(n),yz); fordiv(n,x, x>L & return; fordiv(yz=n/x,y, y>x & break; y*(x+y-1)==yz & return(1)))} \\ _M. F. Hasler_, Nov 07 2010
%K nonn
%O 1,2
%A _Georgi Guninski_, Oct 23 2010
%E More terms from _Robert G. Wilson v_, Oct 25 2010
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