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

1,2

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

Links

Michael De Vlieger, Table of n, a(n) for n = 1..3106 (rows n = 1..18, flattened), first 1502 terms (rows n = 1..15) by T. D. Noe.

Michael De Vlieger, Log log scatterplot of a(n), n = 1..3106, with a color function showing 1 in black, primes in red, proper prime powers in gold, squarefree composites in green, powerful numbers that are not prime powers in purple, and numbers that are neither powerful nor squarefree in blue.

Example

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

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 *)

Crossrefs

Cf. A138180.

Sequence in context: A210688 A328428 A195794 * A275708 A111809 A100869

Adjacent sequences: A145602 A145603 A145604 * A145606 A145607 A145608

Keyword

nonn,tabf

Author

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

Status

approved