A138180 Irregular triangle read by rows: row n consists of all numbers x such that x and x+1 have no prime factor larger than prime(n).

1, 1, 2, 3, 8, 1, 2, 3, 4, 5, 8, 9, 15, 24, 80, 1, 2, 3, 4, 5, 6, 7, 8, 9, 14, 15, 20, 24, 27, 35, 48, 49, 63, 80, 125, 224, 2400, 4374, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 14, 15, 20, 21, 24, 27, 32, 35, 44, 48, 49, 54, 55, 63, 80, 98, 99, 120, 125, 175, 224, 242, 384, 440, 539

Offset: 1

Author

T. D. Noe, Mar 04 2008

Keywords

• nonn
• tabf

Comments

A number x is p-smooth if all prime factors of x are <= p. The length of row n is A002071(n). Row n begins with 1 and ends with A002072(n). Each term of row n-1 is in row n.

The n-th row is the union of the rows 1 to n of A145605. - M. F. Hasler, Jan 18 2015

Examples

			The table reads:
1,
1, 2, 3, 8,
1, 2, 3, 4, 5, 8, 9, 15, 24, 80,  (= A085152)
1, 2, 3, 4, 5, 6, 7, 8, 9, 14, 15, 20, 24, 27, 35, 48, 49, 63, 80, 125, 224, 2400, 4374, (= A085153)
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 14, 15, 20, 21, 24, 27, 32, 35, 44, 48, 49, 54, 55, 63, 80, 98, 99, 120, 125, 175, 224, 242, 384, 440, 539, 2400, 3024, 4374, 9800 (= A252494),
...
		

References

• See A002071.

Links

• T. D. Noe, Rows n=1..10 of triangle, flattened

Crossrefs

Cf. A145605; A085152, A085153, A252494, A252493, A252492.

Programs

Mathematica

(* This program needs x maxima taken from A002072. *) xMaxima = A002072; smoothNumbers[p_, 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]; row[n_] := Module[{sn}, sn = smoothNumbers[Prime[n], xMaxima[[n]]+1]; Reap[Do[If[sn[[i]]+1 == sn[[i+1]], Sow[sn[[i]]]], {i, 1, Length[sn]-1}]][[2, 1]]]; Table[Print[n]; row[n], {n, 1, 10}] // Flatten (* Jean-François Alcover, Jan 16 2015, updated Nov 10 2016 *)

PARI

A138180_row=[]; A138180(n,k)={if(k, A138180(n)[k], #A138180_rowA138180_row=concat(A138180_row,vector(n)); if(#A138180_row[n], A138180_row[n], k=0; p=prime(n); A138180_row[n]=vector(A002071(n),i, until( vecmax(factor(k++)[, 1])<=p && vecmax(factor(k--+(k<2))[, 1])<=p,k++); k)))} \\ A138180(n) (w/o 2nd arg. k) returns the whole row. - M. F. Hasler, Jan 16 2015