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A338059 The Enots Wolley sequence A336957 with the missing prime powers interpolated.

%I A338059 #39 Dec 21 2024 23:51:41
%S A338059 1,2,4,6,3,9,15,5,25,35,7,14,8,12,27,33,11,55,10,16,18,21,49,77,22,20,
%T A338059 45,39,13,26,28,63,51,17,34,32,38,19,57,69,23,46,40,65,91,42,30,85,
%U A338059 119,56,24,75,95,76,36,81,87,29,145,50,44,99,93,31,62,52,117,105,70,58,261,111,37
%N A338059 The Enots Wolley sequence A336957 with the missing prime powers interpolated.
%C A338059 There is a strong conjecture that A336957 consists exactly of 1, 2, and all numbers with at least two different prime factors. [The only uncertainty is whether all numbers with at least two prime factors appear.]
%C A338059 The terms in A000961 greater than 2 are definitely missing from A336957, so A336957 is obviously not a permutation of the positive integers.
%C A338059 The present sequence is obtained by inserting the missing prime powers q = p^k, p >= 2, k >= 1, in their natural positions. More precisely, let the terms of A336957 be [W(i), i >= 1].
%C A338059 Between W(i) and W(i+1) we insert, in order, any prime powers q < W(i+1) which are not yet in the new sequence and satisfy gcd(q, W(i)) > 1 and gcd(q, W(i-1)) = 1.
%C A338059 It is conjectured that this is a permutation of the positive integers.
%H A338059 N. J. A. Sloane, <a href="/A338059/b338059.txt">Table of n, a(n) for n = 1..5000</a>
%e A338059 Suppose n = 4.
%e A338059 The first 5 terms of A336957 are 1,2,6,15,35. The first 7 terms of the present sequence are 1, 2, 4, 6, 3, 9, 15. To see what comes after a(7) = W(4) = 15, we look at the missing prime powers less than W(5) = 35, which are 5, 7, 8, 11, 13, 16, 17, 19, 23, 25, 27, 29, 31. Just two terms, 5 and 25, have a common factor with 15 and are relatively prime to W(3) = 6, so they are adjoined to the sequence.
%e A338059 In short, we adjoin any missing prime powers which are less than W(n+1), have a common factor with W(n), and are relatively prime to W(n-1). We insert them immediately after W(n).
%Y A338059 Cf. A000961, A336957, A338060 (inverse).
%K A338059 nonn
%O A338059 1,2
%A A338059 _Scott R. Shannon_ and _N. J. A. Sloane_, Oct 17 2020