A354755 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 sum a(n) + a(n-1) is distinct from all previous sums a(i) + a(i-1), i=2..n-1.

1, 2, 2, 4, 4, 6, 3, 9, 6, 8, 8, 10, 10, 12, 9, 15, 10, 16, 12, 15, 15, 18, 14, 20, 15, 21, 18, 20, 20, 22, 22, 24, 21, 27, 24, 26, 26, 28, 21, 35, 20, 38, 19, 57, 3, 60, 2, 62, 4, 64, 6, 63, 9, 66, 8, 70, 7, 77, 11, 88, 2, 78, 3, 84, 2, 80, 5, 60, 32, 62, 31, 93, 3, 99, 6, 92, 8, 96, 10, 85, 25

Offset

1,2

Comments

In the first 500000 terms the fixed points are 1,2,4,6,2388,2390,2392,2394; it is likely no more exist. In the same range many numbers do not appear, the lowest five being 59,67,73,89,97. It is possible these and many other numbers never appear although this is unknown.

Links

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Michael De Vlieger, Annotated log-log scatterplot of a(n) n = 1..2^14, showing records in red and a(n) = 2 in blue, highlighting fixed points in gold.

Scott R. Shannon, Image of the first 500000 terms. The green line is y = n.

Example

a(7) = 3 as a(6) = 6, and 3 is the smallest number that shares a factor with 6 and whose sum with the previous term, 6 + 3 = 9, has not appeared. Note 2 shares a factor with 6 but 6 + 2 = 8, and a sum of 8 has already occurred with a(4) + a(5) = 4 + 4 = 8, so 2 cannot be chosen.

Mathematica

nn = 120; c[_] = 0; a[1] = c[1] = 1; a[2] = j = 2; c[3] = 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 *)

Prog

(PARI) lista(nn) = my(va = vector(nn), vs = 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]), vs), k++); va[n] = k; vs[n-2] = k+va[n-1]; ); va; \\ Michel Marcus, Jun 15 2022

Crossrefs

Cf. A064413, A354727, A354687, A354753, A353989, A354087, A352763

Sequence in context: A278233 A354753 A377225 * A354858 A128923 A153835

Adjacent sequences: A354752 A354753 A354754 * A354756 A354757 A354758

Keyword

nonn,look

Author

Scott R. Shannon, Jun 06 2022

Status

approved