A356429 Smallest m such that A356428(m) = n, or -1 if there is no such m.

2, 8, 48, 315, 320, 6664, 135450, 273000, 518661, 519440, 519622, 148830266, 558797841, 558797968, 24900609294

Offset: 1

Author

Jianing Song, Aug 07 2022

Keywords

• nonn
• hard
• more

Comments

a(n) is the smallest m such that there are exactly n distinct gpf(x)'s in the iterations x -> x - gpf(x) starting at m and ending at 0, where gpf = A006530.

Conjecture: a(n) != -1 for all n. This would be true if A356428 is unbounded; otherwise, this sequence consists of entirely -1's after some point.

Since A356428(n) - A356428(n-gpf(n)) = 0 or 1, sequence is strictly increasing if no term equals -1.

If a(m) > -1 for m >= 15 then a(m) > 10^9. - David A. Corneth, Aug 09 2022

Examples

			In the following examples the numbers produced by the iterations are listed together with their GPFs.
320 (5) -> 315 (7) -> 308 (11) -> 297 (11) -> 286 (13) -> 273 (13) -> 260 (13) -> 247 (19) -> ... -> 19 (19) -> 0, the distinct gpf(x)'s are 5, 7, 11, 13, and 19. 320 is the smallest number such that the distinct gpf(x)'s in the iterations is 5, so a(5) = 320.
6664 (17) -> 6647 (23) -> 6624 (23) -> 6601 (41) -> 6560 (41) -> 6519 (53) -> 6466 (53) -> 6413 (53) -> 6360 (53) -> 6307 (53) -> 6254 (59) -> 6195 (59) -> 6136 (59) -> 6077 (103) -> ... -> 103 (103) -> 0, the distinct gpf(x)'s are 17, 23, 41, 53, 59, and 103. 6664 is the smallest number such that the distinct gpf(x)'s in the iterations is 6, so a(6) = 6664.
		

Crossrefs

Cf. A006530 (gpf), A076563, A309892, A356428.

Extensions

a(12) from Michael S. Branicky, Aug 08 2022

a(13)-a(14) from David A. Corneth, Aug 09 2022

a(15) from Jinyuan Wang, Jul 07 2025