A351101 Variation of the Sisyphus sequence A350877: the same rules apply except that each time a(n) is divided by a prime the dividing prime is incremented to the next prime while the prime being added to each term is reset to 2.

1, 3, 6, 3, 5, 8, 13, 20, 31, 44, 61, 80, 103, 132, 44, 46, 49, 54, 61, 72, 85, 17, 19, 22, 27, 34, 45, 58, 75, 94, 117, 146, 177, 214, 255, 298, 345, 398, 457, 518, 74, 76, 79, 84, 91, 102, 115, 132, 12, 14, 17, 22, 29, 40, 53, 70, 89, 112, 141, 172, 209, 250, 293, 340, 393, 452, 513, 580, 651

Offset: 1

Author

Scott R. Shannon, Jan 31 2022

Keywords

• nonn

Comments

This sequence uses the same rules as the Sisyphus sequence, A350877, except that here, instead of always dividing by 2 whenever a(n) is divisible by 2, the prime that is acting as the divisor of a(n), initially 2, is incremented to the next prime once one or more divisions of a(n) by the current dividing prime occur. Additionally each time the dividing prime is incremented the prime being added to each term when a(n) is not divisible by the dividing prime is reset to 2. Once the dividing prime is incremented the terms are then checked for divisibility by this new prime. See the examples below.

In the first 100 million terms the values return to 1 twelve times, see A351278. It is likely this occurs infinitely often although this is unknown. In the same range the smallest number not seen is 5272. This suggests all numbers are eventually visited but this is also unknown. The first numbers to repeat are 3, 44, 61, 132, 17, 22, 75, ... .

In the first 100 million terms the longest gap between prime divisions is 299001 terms, ending at a(80477537) = 609823139121, which is divisible by 40471. Surprisingly the shortest possible gap between different dividing primes, two terms, does not occur until a(194517); the previous terms are a(194514) = 5421744, which is divisible by the dividing prime 1637, a(194515) = 5421744/1637 = 3312, a(194516) = a(194515) + 2 = 3314. The dividing prime is now 1657, and 3314/1657 = 2 = a(194517).

Examples

			a(3) = 6 as a(2) = 3, which is not divisible by the current dividing prime 2, and the next additive prime is 3, so a(3) = 3 + 3 = 6.
a(4) = 3 as a(3) = 6, the current dividing prime is 2, and 6/2 = 3. As 3 is not divisible by 2, the divisions by 2 stop, and the dividing prime becomes 3 while the additive prime resets to 2.
a(5) = 5 as a(4) = 3 and the additive prime is 2, so a(5) = 3 + 2 = 5.
a(6) = 8 as a(5) = 5, which is not divisible by 3, and the next additive prime is 3, so a(6) = 5 + 3 = 8.
a(15) = 44 as a(43) = 132, the current dividing prime is 3, and 132/3 = 44. As 44 is not divisible by 3, the divisions by 3 stop, the dividing prime becomes 5 and the additive prime resets to 2.
		

Links

• Scott R. Shannon, Image of the first 10 million terms.

Crossrefs

Cf. A350877, A351278, A351102, A000040.