A351743 -id:A351743 - cp's OEIS frontend

A355149 Partial sums of A351743.

1, 2, 4, 5, 10, 12, 15, 20, 24, 27, 28, 56, 57, 114, 116, 145, 150, 156, 169, 170, 340, 341, 682, 684, 855, 860, 1032, 1035, 1150, 1152, 1161, 1204, 1232, 1243, 1356, 1368, 1387, 1460, 1480, 1517, 1558, 1596, 1617, 1628, 1776, 1779, 2372, 2376, 2673, 2674

Offset: 1

Michael De Vlieger, Rémy Sigrist and David James Sycamore, Jun 21 2022

nonn

a(n) is odd iff n mod 3 = 1. Parity of a(n) is related to that of A351743(n).

			a(3) = A351743(1) + A351743(2) + A351743(3) = 1 + 1 + 2 = 4.

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

Cf. A351743.

nn = 50, a[1] = m = s = 1; Do[If[m == 1, Set[k, s], Set[k, s]; While[GCD[m, k] > 1, k /= GCD[m, k]]]; m = k; s += k; a[n] = s, {n, 2, nn}]; Array[a, nn]

{ s=0; for (n=1, 50, if (n==1, v=1, w=s; while (1 < g=gcd(w, v), w /= g); v=w); print1 (s+=v", ")) }

Conjecture: a(n + 42) = 3645*a(n) + b(n) for n >= 138 (where b is 42-periodic).

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.

Original entry on oeis.org

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

David James Sycamore, Oct 11 2022

nonn

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).

			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.

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.

Cf. A064413, A084385, A347113, A351743.

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 *)

cp's OEIS Frontend

A355149 Partial sums of A351743.

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