A357735 a(1)=1, a(2)=2. Thereafter a(n+1) is least k != partial sum s(n) which has not occurred earlier, such that gcd(k, s(n)) > 1.

1, 2, 6, 3, 4, 8, 9, 11, 10, 12, 14, 5, 15, 16, 18, 20, 7, 21, 13, 24, 27, 22, 26, 28, 23, 25, 30, 32, 33, 31, 34, 35, 40, 44, 55, 36, 37, 39, 17, 42, 45, 38, 46, 48, 50, 19, 57, 52, 41, 62, 43, 54, 56, 58, 60, 64, 51, 63, 66, 49, 70, 77, 68, 69, 72, 75, 74, 76

Offset: 1

Author

David James Sycamore, Oct 11 2022

Keywords

• nonn

Comments

It follows from the definition that if s(n) is prime then a(n+1) = 2*s(n). This happens only once in the sequence, when a(3)=6, following s(2)=3. For all n > 2 s(n) is composite. Conjectured to be a permutation of the positive integers (primes not in natural order).

Examples

			Since a(1)=1 and a(2)=2, we have s(2)=3, then a(3) is 6, the smallest unused term sharing a divisor with 3.
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 1..16384
• Michael De Vlieger, Log-log scatterplot of a(n) n = 1..2^17, labeling records in red, local minima in blue, and highlighting prime terms in green.

Crossrefs

Cf. A064413, A084385, A347113, A351743.

Programs

Mathematica

nn = 2^16; c[] = False; Array[Set[{a[#], c[#]}, {#, True}] &, 2]; u = s = 3; Do[k = u; While[Nand[! c[k], ! CoprimeQ[k, s], k != s], k++]; Set[{a[n], c[k]}, {k, True}]; s += k; If[k == u, While[c[u], u++]], {n, 3, nn}]; Array[a, nn] (* _Michael De Vlieger, Oct 11 2022 *)