A145605 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.

1, 2, 3, 8, 4, 5, 9, 15, 24, 80, 6, 7, 14, 20, 27, 35, 48, 49, 63, 125, 224, 2400, 4374, 10, 11, 21, 32, 44, 54, 55, 98, 99, 120, 175, 242, 384, 440, 539, 3024, 9800, 12, 13, 25, 26, 39, 64, 65, 77, 90, 104, 143, 168, 195, 324, 350, 351, 363, 624, 675, 728, 1000, 1715

Offset: 1

Author

T. D. Noe, Oct 14 2008, Nov 03 2008

Keywords

• nonn
• tabf

Comments

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.

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

Examples

			1
2, 3, 8
4, 5, 9, 15, 24, 80
6, 7, 14, 20, 27, 35, 48, 49, 63, 125, 224, 2400, 4374
		

Links

• T. D. Noe, Rows n=1..15 of triangle, flattened

Programs

Mathematica

(* 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 *)