A101513 a(1) = 1, a(2) = 2, a(3) = 3; triangle where n-th row has lowest n positive integers not yet in the sequence such that each integer has a prime divisor in common with at least one element of the (n-1)th row.

1, 2, 3, 4, 6, 8, 9, 10, 12, 14, 5, 7, 15, 16, 18, 20, 21, 22, 24, 25, 26, 11, 13, 27, 28, 30, 32, 33, 34, 35, 36, 38, 39, 40, 42, 44, 17, 19, 45, 46, 48, 49, 50, 51, 52, 23, 54, 55, 56, 57, 58, 60, 62, 63, 64, 29, 31, 65, 66, 68, 69, 70, 72, 74, 75, 76, 37, 77, 78, 80, 81, 82

Offset

1,2

Comments

Is this a permutation of the positive integers?

Conjectures from N. J. A. Sloane, Apr 22 2005: (Start)

"Call a number "postponed" if it cannot be placed right away, that is, if it is relatively prime to the numbers in the previous row. Then I conjecture that:

"(1) a number n >= 4 is postponed iff n is prime,

"(2) every number appears,

"(3) the primes appear in order,

"(4) 2p (p prime) will appear in one row and p will appear in the next row,

"(5) let c(i) = A018252(i) be the i-th nonprime and define a sequence k(n) [see A104655], n >= 3, by k(3) = 4 and for n >= 4, n*(n+1)/2 = pi( floor( c(k(n-1))/2 ) ) + k(n). Then the final term in row n, for n >= 3, is c(k(n)) [A104656]." (End)

Links

Michael De Vlieger, Table of n, a(n) for n = 1..11628 (rows 1 <= n <= 150).

Example

Triangle begins:
   1;
   2,  3;
   4,  6,  8;
   9, 10, 12, 14;
   5,  7, 15, 16, 18;
  20, 21, 22, 24, 25, 26;
  11, 13, 27, 28, 30, 32, 33;
  ...
7 is in the 5th row because it does not occur earlier and 14 is in the 4th row.

Mathematica

f[w_List] := Block[{k = 4, m = {}}, Do[While[Nand[FreeQ[Join[w, m], k], AnyTrue[Last@ w, GCD[k, #] > 1 &]], k++]; AppendTo[m, k], {i, Length@ w + 1}]; m]; Nest[Append[#, f@ #] &, Table[n + k - 1, {n, 2}, {k, n}], 10] // Flatten (* Michael De Vlieger, Sep 25 2017 *)

Crossrefs

Cf. the EKG sequence A064413. See also A104654, A104655, A104656.

Sequence in context: A015858 A321473 A125292 * A153276 A298866 A108408

Adjacent sequences: A101510 A101511 A101512 * A101514 A101515 A101516

Keyword

nonn,tabl

Author

Leroy Quet, Jan 25 2005

Extensions

More terms from Joshua Zucker, May 20 2006

Status

approved