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

1,3

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

References

See A002071.

Links

Michael De Vlieger, Table of n, a(n) for n = 1..13156 (rows n = 1..18, flattened), first 1007 terms (rows n = 1..10) by T. D. Noe.

Michael De Vlieger, Extended table of n, a(n) for n = 1..27931 (rows n = 1..21, flattened).

Michael De Vlieger, Log log scatterplot of a(n), n = 1..13156, 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.

Michael De Vlieger, Plot prime(i)^m | a(n) at (x,y) = (n,i), n = 1..2142, with a color function showing m = 1 in black, m = 2 in red, m = 3 in orange, ..., m = 13 in magenta. The bar below the plot indicates a term a(n) using the color function immediately above.

OEIS Index entries for sequences related to the abc conjecture

Example

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),
...

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 *)

Prog

(PARI) A138180_row=[]; A138180(n, k)={if(k, A138180(n)[k], #A138180_row<n && A138180_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

Crossrefs

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

Sequence in context: A131959 A202688 A021046 * A345093 A079585 A354854

Adjacent sequences: A138177 A138178 A138179 * A138181 A138182 A138183

Keyword

nonn,tabf

Author

T. D. Noe, Mar 04 2008

Status

approved