A101513 - OEIS

%I #18 Jan 22 2022 08:46:43

%S 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,

%T 32,33,34,35,36,38,39,40,42,44,17,19,45,46,48,49,50,51,52,23,54,55,56,

%U 57,58,60,62,63,64,29,31,65,66,68,69,70,72,74,75,76,37,77,78,80,81,82

%N 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.

%C Is this a permutation of the positive integers?

%C Conjectures from _N. J. A. Sloane_, Apr 22 2005: (Start)

%C "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:

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

%C "(2) every number appears,

%C "(3) the primes appear in order,

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

%C "(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)

%H Michael De Vlieger, <a href="/A101513/b101513.txt">Table of n, a(n) for n = 1..11628</a> (rows 1 <= n <= 150).

%e Triangle begins:

%e    1;

%e    2,  3;

%e    4,  6,  8;

%e    9, 10, 12, 14;

%e    5,  7, 15, 16, 18;

%e   20, 21, 22, 24, 25, 26;

%e   11, 13, 27, 28, 30, 32, 33;

%e   ...

%e 7 is in the 5th row because it does not occur earlier and 14 is in the 4th row.

%t 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 *)

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

%K nonn,tabl

%O 1,2

%A _Leroy Quet_, Jan 25 2005

%E More terms from _Joshua Zucker_, May 20 2006