cp's OEIS Frontend

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.

%I A101513 #18 Jan 22 2022 08:46:43
%S A101513 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 A101513 32,33,34,35,36,38,39,40,42,44,17,19,45,46,48,49,50,51,52,23,54,55,56,
%U A101513 57,58,60,62,63,64,29,31,65,66,68,69,70,72,74,75,76,37,77,78,80,81,82
%N 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.
%C A101513 Is this a permutation of the positive integers?
%C A101513 Conjectures from _N. J. A. Sloane_, Apr 22 2005: (Start)
%C A101513 "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 A101513 "(1) a number n >= 4 is postponed iff n is prime,
%C A101513 "(2) every number appears,
%C A101513 "(3) the primes appear in order,
%C A101513 "(4) 2p (p prime) will appear in one row and p will appear in the next row,
%C A101513 "(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 A101513 Michael De Vlieger, <a href="/A101513/b101513.txt">Table of n, a(n) for n = 1..11628</a> (rows 1 <= n <= 150).
%e A101513 Triangle begins:
%e A101513    1;
%e A101513    2,  3;
%e A101513    4,  6,  8;
%e A101513    9, 10, 12, 14;
%e A101513    5,  7, 15, 16, 18;
%e A101513   20, 21, 22, 24, 25, 26;
%e A101513   11, 13, 27, 28, 30, 32, 33;
%e A101513   ...
%e A101513 7 is in the 5th row because it does not occur earlier and 14 is in the 4th row.
%t A101513 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 A101513 Cf. the EKG sequence A064413. See also A104654, A104655, A104656.
%K A101513 nonn,tabl
%O A101513 1,2
%A A101513 _Leroy Quet_, Jan 25 2005
%E A101513 More terms from _Joshua Zucker_, May 20 2006