Post #2658771

2026-05-06 05:02 UTC

Lay down number bars 1, 2, 3, ... Put the bar as far to the left as you can, as high as you can: if you can stack on top of previous bars, then do so. The first 25 bars look like the attached image. The sequence is the lengths of the leftmost bar on each row. Next two terms are 32, 40. Now in the #OEIS: https://oeis.org/A395531

Replies (5)

@ngons@mathstodon.xyz 2026-05-06 05:57

@christianp@mathstodon.xyz That is beautiful! I think the lower row also is a unique sequence 1,2,4,5,8,12,17,19,20 has no hits in OEIS

Open ##2658772
@hellman@mathstodon.xyz 2026-05-06 07:59

@christianp@mathstodon.xyz Most of the time a(n)=a(n-1)+n, so can be "compressed" by looking into those n where it does not hold. What is interesting, quite many of those actually belong to A395531 itself! ``` n = 3, in A395531? yes, a(n)-a(n-1) = 4 = n + 1 n = 6, in A395531? - , a(n)-a(n-1) = 9 = n + 3 n = 11, in A395531? yes, a(n)-a(n-1) = 14 = n + 3 n = 16, in A395531? yes, a(n)-a(n-1) = 20 = n + 4 n = 22, in A395531? - , a(n)-a(n-1) = 30 = n + 8 n = 32, in A395531? yes, a(n)-a(n-1) = 38 = n + 6 n = 40, in A395531? yes, a(n)-a(n-1) = 47 = n + 7 n = 49, in A395531? yes, a(n)-a(n-1) = 57 = n + 8 n = 59, in A395531? yes, a(n)-a(n-1) = 68 = n + 9 n = 67, in A395531? - , a(n)-a(n-1) = 70 = n + 3 n = 70, in A395531? - , a(n)-a(n-1) = 80 = n + 10 n = 85, in A395531? yes, a(n)-a(n-1) = 96 = n + 11 n = 98, in A395531? yes, a(n)-a(n-1) = 110 = n + 12 ``` https://gist.github.com/hellman/a4a56f013085edf379ac6377db0fa9bb

Open ##2658773
@robinhouston@mathstodon.xyz 2026-05-06 08:13

@christianp@mathstodon.xyz What a shame the name ‘Perfect numbers’ is already taken.

Open ##2658778
@robinhouston@mathstodon.xyz 2026-05-06 08:49

@christianp@mathstodon.xyz This process is so interesting! Does this line look pretty straight to you? (Number of bars per row once 10 million bars have been put in.)

Open ##2658780
@zimpenfish@social.rjp.is 2026-05-06 11:56

@christianp@mathstodon.xyz Probably come up already but the deltas seem to make a nice "diagonal with bumps" pattern. Illustrated here via the magic of awk. (I imagine there's an obvious explanation for this but I am 0% mathematician.)

Open ##2658784