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%I #16 Mar 16 2019 18:35:09
%S 1,2,6,3,12,4,20,5,10,14,7,21,15,24,8,40,30,18,9,36,28,35,42,22,11,33,
%T 39,13,26,34,17,51,48,16,80,45,55,44,52,60,57,19,38,46,23,69,66,50,25,
%U 75,78,54,27,108,68,76,84,56,63,70,58,29,87,93,31,62,74
%N Lexicographically earliest sequence of distinct positive numbers such that, for any n > 1, the prime factorizations of a(n) and of a(n+1) share a prime power > 1.
%C For any n > 1, there is a prime number p that divides a(n) and such that the p-adic valuation of a(n) equals the p-adic valuation of a(n+1).
%C This sequence has similarities with the EKG sequence A064413: here consecutive terms share a prime power, there consecutive terms share a prime factor.
%C The scatterplots of this sequence and of A064413 show three similar lines starting from the origin; however here we also have some isolated points.
%C This sequence and A064413 match for n = 1, 2, 9, 28, 78, 113, 245, 419, 420, 421, 580, 581, 582, 644, 816, 937, 969, 1105, 1117, 1118, etc.
%C This sequence is likely to be a permutation of the natural numbers.
%H Rémy Sigrist, <a href="/A297076/b297076.txt">Table of n, a(n) for n = 1..10000</a>
%H Rémy Sigrist, <a href="/A297076/a297076.gp.txt">PARI program for A297076</a>
%e The first terms, alongside their prime factorizations, are:
%e n a(n) Prime factorization of a(n)
%e -- ---- ---------------------------
%e 1 1 1
%e 2 2 2
%e 3 6 2 * 3
%e 4 3 3
%e 5 12 2^2 * 3
%e 6 4 2^2
%e 7 20 2^2 * 5
%e 8 5 5
%e 9 10 2 * 5
%e 10 14 2 * 7
%e 11 7 7
%e 12 21 3 * 7
%e 13 15 3 * 5
%e 14 24 2^3 * 3
%e 15 8 2^3
%e 16 40 2^3 * 5
%e 17 30 2 * 3 * 5
%e 18 18 2 * 3^2
%e 19 9 3^2
%e 20 36 2^2 * 3^2
%o (PARI) See Links section.
%Y Cf. A064413.
%K nonn
%O 1,2
%A _Rémy Sigrist_, Dec 25 2017