A356428 -id:A356428 - cp's OEIS frontend

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

Jianing Song, Aug 07 2022

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

			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.

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

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

A356438 Numbers k such that A309892(k) = k/gpf(k), where gpf = A006530.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 9, 10, 11, 12, 13, 14, 15, 17, 19, 20, 21, 22, 23, 25, 26, 28, 29, 30, 31, 33, 34, 35, 37, 38, 39, 41, 42, 43, 44, 46, 47, 49, 51, 52, 53, 55, 56, 57, 58, 59, 61, 62, 63, 65, 66, 67, 68, 69, 70, 71, 73, 74, 76, 77, 78, 79, 82, 83, 85

Offset: 1

Jianing Song, Aug 07 2022

Note that A309892(k) <= k/gpf(k); these sequence lists k such that the equality holds.

For k >= 2, k is a term if and only if k/gpf(k) < nextprime(gpf(k)), where nextprime = A151800.

			15 is a term since the number of steps needed to reach 0 of the iteration x -> x - gpf(x) starting at 15 is 3: 15 -> 10 -> 5 -> 0, and 3 = 15/gpf(15).

Jianing Song, Table of n, a(n) for n = 1..36902 (all terms <= 50000)

Other than 1, indices of 1 in A356428.
Includes A000040 and A001358 as subsequences.
Complement of A356441.
Cf. A309892, A006530 (gpf), A151800, A076563.

isA356438(n) = if(n>1, my(p=vecmax(factor(n)[, 1])); n/p

cp's OEIS Frontend

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

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