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%I #12 Nov 17 2016 08:00:28
%S 1,2,3,8,4,5,9,15,24,80,6,7,14,20,27,35,48,49,63,125,224,2400,4374,10,
%T 11,21,32,44,54,55,98,99,120,175,242,384,440,539,3024,9800,12,13,25,
%U 26,39,64,65,77,90,104,143,168,195,324,350,351,363,624,675,728,1000,1715
%N Irregular triangle in which row n consists of all numbers x such that x and x+1 are both prime(n)-smooth numbers but not both prime(n-1)-smooth.
%C The length of row n is A145604(n). The largest x in row n is A145606(n). This is sequence A138180 with only the first occurrence of each number retained. Row n begins with prime(n)-1.
%C A permutation of the positive integers (when seen as linear sequence). A252489(n) yields the row in which n appears in the table. - _M. F. Hasler_, Jan 16 2015
%H T. D. Noe, <a href="/A145605/b145605.txt">Rows n=1..15 of triangle, flattened</a>
%e 1
%e 2, 3, 8
%e 4, 5, 9, 15, 24, 80
%e 6, 7, 14, 20, 27, 35, 48, 49, 63, 125, 224, 2400, 4374
%t (* Computation using x maxima taken from A145606 *) A145606 = {1, 8, 80, 4374, 9800, 123200, 336140, 11859210, 5142500, 177182720, 1611308699, 3463199999, 63927525375, 421138799639, 1109496723125}; smoothNumbers[p_?PrimeQ, max_] := Module[{a, aa, k, pp, iter}, k = PrimePi[p]; aa = Array[a, k]; pp = Prime[Range[k]]; iter = Table[{a[j], 0, PowerExpand @ Log[pp[[j]], max/Times @@ (Take[pp, j - 1]^Take[aa, j - 1])]}, {j, 1, k}]; Table[Times @@ (pp^aa), Sequence @@ iter // Evaluate] // Flatten // Sort]; smoothMax[n_] := A145606[[n]]; row[n_] := Module[{sn, sn1}, sn = smoothNumbers[Prime[n], smoothMax[n] + 1] ; sn1 = smoothNumbers[Prime[n - 1], smoothMax[n] + 1] ; Select[sn, MemberQ[sn, # + 1] && Not[MemberQ[sn1, #] && MemberQ[sn1, # + 1]] &]]; row[1] = {1}; Table[ro = row[n]; Print[n, " ", ro // Short]; ro, {n, 1, 10}] // Flatten (* _Jean-François Alcover_, Nov 17 2016 *)
%K nonn,tabf
%O 1,2
%A _T. D. Noe_, Oct 14 2008, Nov 03 2008
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