The On-Line Encyclopedia of Integer Sequences (OEIS)

Revisions by Kevin Ryde (See also Kevin Ryde's wiki page
and changes approved by Kevin Ryde)

(Bold, blue-underlined text is an addition; faded, red-underlined text is a ~~deletion~~.)
Showing entries 1-10 | older changes

A261870

a(n) = minimal number of steps to get from n to 1, where a step is x -> 3x-1 if x is odd, or x -> either x/2 or 3x-1 if x is even.
(history; published version)

#11 by Kevin Ryde at Mon Dec 02 01:23:48 EST 2024

STATUS

editing

proposed

Discussion

Mon Dec 02

07:54

Peter Luschny: Impressive code.

#10 by Kevin Ryde at Mon Dec 02 01:22:18 EST 2024

LINKS

Max Alekseyev, Kevin Ryde, <a href="/A261870/b261870_1.txt">Table of n, a(n) for n = 1..10000</a> (terms 1..1000</a> from Max Alekseyev)

Kevin Ryde, <a href="/A261870/a261870.c.txt">C Code</a>

PROG

(C) /* See links. */

STATUS

approved

editing

A378569

allocated for Vincenzo Librandi
(history; published version)

#15 by Kevin Ryde at Sun Dec 01 20:34:40 EST 2024

STATUS

editing

proposed

#14 by Kevin Ryde at Sun Dec 01 20:32:20 EST 2024

NAME

Irregular triangle whose n-th read by rows where row = n lists the divisors( of prime(4*n+8) + 2*n + 3).

COMMENTS

Irregular triangle

EXAMPLE

k=1 2 3 4 5 6 7 8

[ n=1: 1, 2, 3, 6, 7, 14, 21, 42], (A018258).

[ n=2: 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60], (A018266).

...

STATUS

proposed

editing

Discussion

Sun Dec 01

20:34

Kevin Ryde: I think it always helps to have example showing row numbers for quick reference which is which.  Somebody one day will want to know the "k" column numbering too.

A378119

allocated for Barney Maunder-Taylor
(history; published version)

#11 by Kevin Ryde at Sun Dec 01 20:21:55 EST 2024

STATUS

editing

proposed

#10 by Kevin Ryde at Sun Dec 01 20:21:07 EST 2024

COMMENTS

It is clear that a(n) must be zero if n is a multiple of three 3 since if we assume (hoping for a contradiction) that there exist k, k+1 both of which have digit sums that are multiples of 3, then k and k+1 must themselves both be divisible by 3, a contradiction.

Discussion

Sun Dec 01

20:21

Kevin Ryde: The word from on high has usually been to prefer -1 as a "no such exists" marker.  (Absent a reason like helping a formula or something.)

#9 by Kevin Ryde at Sun Dec 01 20:16:18 EST 2024

NAME

a(n) = the smallest positive k such that the digit sums of k and k + 1 are both divisible by n, or 0 if no such pair k exists.

STATUS

proposed

editing

A378559

allocated for Tamas Sandor Nagy
(history; published version)

#23 by Kevin Ryde at Sun Dec 01 17:23:20 EST 2024

STATUS

editing

proposed

Discussion

Mon Dec 02

00:13

Tamas Sandor Nagy: @Kevin -- Thank you for the editing. The name is now more succint and mathematical.
Yes, it occured to me too that a(n) may not always exist. But I thought that unlikely because the full set of remainders for p tends to be only a tiny front-portion of the set corresponding to t. Still, I could put that question in the Comments since you suggested it.
Originally, I thought of making an array of chains like 3 -> 5 -> 17 -> 2162177 -> ... ?
 and 7 -> 67 ->  23645337682056445350227232067 -> ... ?,  etc., but computing such vast numbers was an obstacle for me. That question can be extended to asking, are all these chains infinite or some are broken at some point?

04:07

Amiram Eldar: According to Dirichlet's theorem on arithmetic progressions a(n) exists for all n. This is because  A058254(n) and prime(n) are coprime, so there are infinitely many primes of the form prime(n) + k* A058254(n).

05:42

Tamas Sandor Nagy: @Amiram -- Thank you for clarifying this point.

#22 by Kevin Ryde at Sun Dec 01 17:21:01 EST 2024

EXAMPLE

a(4) = 67 because prime(4) = 7 , and 7 and 67 mod each k = {2, 3, 4, 5, 6} both have remainders {1, 1, 3, 2, 1} in that order, and 67 is the smallest prime after 7 that has this property.

Discussion

Sun Dec 01

17:22

Kevin Ryde: Think express by congruence rather than remainders.  Yes that sort of LCM.  I started with a chinese remainder theorem for what those simultaneous congruences become, but it's easier than that.

17:23

Kevin Ryde: It there any question whether a(n) exists?  Something about primes in arithmetic progression, or explicitly make something which has to be prime by big enough multiple?

#21 by Kevin Ryde at Sun Dec 01 17:19:25 EST 2024

NAME

a(n) = t is the smallest prime, other than the n-th t > p = prime p, that gives the same remainder (n) for each k as which t == p does when divided by (mod k) for all k = (2,...(,p - 1)).

EXAMPLE

a(4) = 67 because the 4th prime is (4) = 7; when both and 7 and 67 are divided by mod k = {2, 3, 4, 5 and , 6, the } both have remainders for both are {1, 1, 3, 2 and , 1 } in that order, and 67 is the smallest prime after 7 that has this property.

STATUS

proposed

editing