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A354753
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a(1) = 1, a(2) = 2; for n > 2, a(n) is the smallest positive number that shares a factor with a(n-1) and the product a(n) * a(n-1) is distinct from all previous products a(i) * a(i-1), i=2..n-1.
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10
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1, 2, 2, 4, 4, 6, 2, 10, 4, 8, 6, 3, 3, 9, 6, 6, 10, 5, 5, 15, 3, 21, 6, 12, 8, 8, 10, 10, 12, 9, 9, 15, 6, 14, 2, 22, 4, 14, 7, 7, 21, 9, 18, 8, 14, 10, 15, 12, 14, 14, 16, 8, 20, 10, 22, 6, 26, 2, 34, 4, 26, 8, 22, 11, 11, 33, 3, 39, 6, 32, 8, 30, 9, 24, 12, 21, 14, 20, 15, 15, 21, 18, 18
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OFFSET
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1,2
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COMMENTS
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This sequence uses a similar rule to A088177 but here all neighboring terms also share a factor. In the first 500000 terms the fixed points are 1,2,4,6, it is likely no more exist, while the smallest number not to have appeared is 1153. The sequence is conjectured to be a permutation of the positive integers.
See A354754 for the products of all pairs of terms.
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LINKS
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EXAMPLE
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a(7) = 2 as a(6) = 6 and 2 is the smallest positive number that shares a factor with 6 and whose product with 6, 2 * 6 = 12, has not previously appeared.
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MATHEMATICA
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nn = 120; c[_] = 0; a[1] = c[1] = 1; a[2] = a[2] = j = 2; Do[k = 2; While[Nand[c[j*k] == 0, ! CoprimeQ[j, k]], k++]; Set[{a[n], c[j*k]}, {k, n}]; j = k, {n, 3, nn}]; Array[a, nn] (* Michael De Vlieger, Jun 15 2022 *)
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PROG
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(PARI) lista(nn) = my(va = vector(nn), vp = vector(nn-2)); va[1] = 1; va[2] = 2; for (n=3, nn, my(k=2); while ((gcd(k, va[n-1]) == 1) || #select(x->(x==k*va[n-1]), vp), k++); va[n] = k; vp[n-2] = k*va[n-1]; ); va; \\ Michel Marcus, Jun 14 2022
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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