Squares and triangles
Hello SeqFans,
It is possible to
write down in a 5 x 5 square (25 cells) the integers from 1 to 17 -- if you use
one digit per cell:
1 2 3 4 5
6 7 8 9 1
0 1 1 1 2
1 3 1 4 1
5 1 6 1 7
The 4 x 4 square is
impossible to fill exactly, using the same constraint:
1 2 3 4
5 6 7 8
9 1 0 1
1 1 2 1 3
We see that 12 leaves an empty cell, and 13 needs one too much.
What are the exact
"square-filling" integers?
I guess S starts:
S= 1, 4, 9, 17, 29, 45, 65, 89, 111, 144, 183, 228,...
The equivalent seq T could be
constructed for exact "right-triangle-filling" integers:
T= 1, 3, 6, 12, 15, 27, 32, 50, 57,...
None of those are in
the
O
N L
I N E
E N C Y
C L O P E
D I A O F I
N T E G E R S
E Q U E N C E S
Best,
É.
- - - - - - - - - - -
-
(What about other
bases?)
__________
[Alois Heinz]:
Dear Eric,
"square-filling":
S = 1, 4, 9, 17, 29, 45, 65, 89, 111, 144, 183, 228, 279, 336, 399, 468,
543, 624, 711, 804, 903, 1033, 1089, 1147, 1207, 1269, 1333, 1399, 1467, 1537,
1609, 1683, 1759, 1837, 1917, 1999, 2083, 2169, 2257, 2347, 2439, 2533, 2629,
2727, 2827, 2929, 3033, 3139, 3247, 3357, 3469, 3583, 3699, 3817, 3937, 4059,
4183, 4309, 4437, 4567, 4699, 4833, 4969, 5107, 5247, 5389, 5533, 5679, 5827,
5977, 6129, 6283, 6439, 6597, 6757, 6919, 7083, 7249, 7417, 7587, 7759, 7933,
8109, 8287, 8467, 8649, 8833, 9019, 9207, 9397, 9589, 9783, 9979,...
"right-triangle-filling":
T = 1, 3, 6, 12, 15, 27, 32, 50, 57, 81, 90, 106, 113, 128, 136, 153,
162, 181, 191, 212, 223, 246, 258, 283, 296, 323, 337, 366, 381, 412, 428, 461,
478, 513, 531, 568, 587, 626, 646, 687, 708, 751, 773, 818, 841, 888, 912, 961,
986, 1047, 1107, 1212, 1278, 1393, 1465, 1590, 1668, 1803, 1887, 2032, 2122,
2277, 2373, 2538, 2640, 2815, 2923, 3108, 3222, 3417, 3537, 3742, 3868, 4083,
4215, 4440, 4578, 4813, 4957, 5202, 5352, 5607, 5763, 6028, 6190, 6465, 6633,
6918, 7092, 7387, 7567, 7872, 8058, 8373, 8565, 8890, 9088, 9423, 9627, 9972, ...
Best regards,
Alois.
__________
[Franklin T. Adams-Watters]:
If this becomes a
sequence, I would recommend changing it to "The integer k such that the
digits from 1 to k have exactly n^2 digits, or zero if this does not
exist." So it would start:
1,4,9,0,17,0,29,...
(I’ve also dropped
the reference to "filling" squares - we’re ultimately just counting
digits here. Filling squares is the idea that got you here, but not the essence
of what you’ve gotten to. You should mention it in a
comment.)
Don’t forget to
cross-reference A058183.
__________
[Neil Sloane]:
I think we should use
both styles for both sequences, so we will have four new sequences in all.
And I do like Eric’s idea
of filling a square or a right triangle. I would like to see that as part
of the definition.
Otherwise we will be
led to consider:
– The integer k such
that the numbers from 1 to k contain exactly A123456(n) digits,
or 0 if no such k exists, where A123456 is any of the core sequences. Mentioning the square in
the definition makes it more interesting.
Best regards
Neil
__________
[Ed Jeffery]:
NJAS>I think we should use both styles
for both sequences, so we will have four new sequences in all.
Also, Eric’s squares
or triangles could be filled with 0,1,2,..., starting
the filling process with 0 instead of 1. In this case, the 4X4 square could
then be filled but not the 5X5 one. What do these sequences look like?
Ed Jeffery
__________
[Giovanni Resta]:
Hi, nice
construction. You may wander which is the smallest number > 1 that, like 1,
fills a square and also a right-triangle.
Well, it is pretty
big. About 6.2 * 10^2986, or more precisely:
6228698895730227622755017731072145749183515783656112726019201287058092
2562127236929760160301727251033583512368516378393917224776085871444518
8825719501371537699313070671017389828517880860707093862732193680784749
6621557518490349437015983228744446188468153887289398882395727535291785
6683181999963013936342650056306361660262151104039645779303510198956180
7285728290229237337211083263093575771122085710895787248657999286557156
5137232813292358737451625129195611681292838697972546195652235523589263
0447300443190231324556886679142046757929592983128212573123948430665100
0708779793996206974927610083558287763387016479083281131264020268599016
6035002366526140699589820374956004851298324590962505971005956879520024
9147516994600018506426574739288387990050844242012465645381490927710903
5767415734314084228260736534470015202446936279034710754766708206228703
3574100371557777066908510977109222041722437886461814447266521562744785
1502774673786216662931431331444212741301495103436025666861393846438693
6002119620652654163848254601321931968883449559248743373522754500223340
4758404556181396890590237349840707750663069582055851031435285765680613
8988975906842844329713639207954685172731984102518544219442059796155464
9118527918773019164745397113454787619453625207352333698362305237458276
5928903072065242851929074431246729666773046363481381201572370388828426
2554234488010050466314177694884194539567678130889007733708736099650498
9115022850191750608244987574274448634432190712037535921319857483986977
3060012604091916275814247608306689502889904404424982077043886241708604
6972127535543973950624249316926758694564298464342979672803756998108940
8434453452970811973494593138551992181572622010040812956169357838675485
9319780665998058621140239545133749576144692485488266513026566086215874
1015823930258465091835488043916551979836107221129434045230895674611927
7887735482195733755249596321807435909049294991542151006229801621768130
6161915962995722174699037535868229569439147467703620632534904527331736
3699367406595387264632059212963778255737905781420405611809842908898112
6015129332732660929052983199722087654854165243203192941140054740139706
6918830808558927336081386371607313235932540040068206166942914778153523
6748138287365048852549684632498524180330920783392721711445616152963533
6283783381063161352768700120603189757857527802528103917952419044340450
8879042141061416732553147762316910295873725809942419246794121066806927
9911711604863764718135022856268645775714946096018346739641618742013487
0291442485771698670169833005161929066287617581929582922321166308640116
9448908953413818679352851622269726917615836037363858335860231638395996
0315734366961989522578300255066791126509312988929436799707703210297246
4865058111240607999731070044173352214887310681704330308411259801160894
9388747161796719506012351250060201212284453690297232438201773649567978
6962681125204324855961813796111585185042444543700862636858311374167407
5213651465745760075698722365786828520220985496293063853507789852332563
00584073589125271701202183677913121040330835520.
__________
[Robert G. Wilson V]:
That is really quite a number, which is not in the OEIS
= 2^6*3^3*5*13*19*7901*C2978.
__________
[Emmanuel Vantieghem]:
Shouldn’t it be 2^6*3^2*5*13*19*7901*C2978
?
__________
[Robert G. Wilson V]:
Yes, I guess that I mistyped. As for C2978, I let ECM
run overnight with no success at another factor.
Bob.
__________
[юрий герасимов]:
Dear Seq Fans,
I propose to extend
Eric’s idea. This idea is called:
Filling the sides of
square or triangle spirals (with digits).
For square spiral:
S(n) = 1, 2, 3, 5, 7,
0, 11, 13, 15, 0, 20, 23, 26, 0, 33, 37, 41, 0, 50, 55, 60, 0, 71, 77, 83, 0,
96, 0, 0, 0, 0, 0, 127,0,...
For triangle spiral:
S(n) = 1, 2, 4, 7,
10, 0, 0, 19, 23, 0, 0, 38, 44, 0, 0, 65, 73, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 0, ....
Without zeros: 1, 2, 4, 7, 19, 23, 38, 44, 65, 73,... What in the next
one?
First differences: 1,
2, 3, 12, 4, 15, 6, 25, 8,... What in the next one?
Primes in S(n): 2, 7, 19, 23,
73,... What in the
next one?
Example:
9,
1, 0,
9, 5, 9,
2, 6, 6, 9,
9, 6, 4, 4, 8,
3, 6, 4, 4, 6, 8,
9, 7, 5, 2, 3, 3, 8,
4, 6, 4, 8, 7, 4, 6, 7,
9, 8, 6, 2, 1, 2, 2, 2,
8,
5, 6, 4, 9, 6, 5, 6, 4, 6,
6,
9, 9, 7, 3, 1, 7, 1, 2, 1,
1, 8,
6, 7, 4, 0, 7, 8, 6, 4, 5,
4, 6, 5,
9, 0, 8, 3, 1, 9, 1, 5, 1, 2,
0, 0, 8,
7, 7, 4, 1, 8, 1, 2, 3, 4, 3,
4, 4, 6, 4,
9, 1, 9, 3, 1, 0, 1, 1, 1, 2,
1, 2, 9, 9, 8,
8, 7, 5, 2, 9, 2, 0, 2, 1, 2, 2,
2, 3, 3, 5, 3,
9, 2, 0, 3, 3, 3, 4, 3, 5, 3, 6,
3, 7, 3, 8, 8, 8,
9, 7, 5, 1, 5, 2, 5, 3, 5, 4, 5,
5, 5, 6, 5, 7, 5, 2,
1, 3, 7, 4, 7, 5, 7, 6, 7, 7, 7, 8,
7, 9, 8, 0, 8, 1, 8,
0, 0, 1, 0, 1, 1, 0, 2, 1, 0, 3, ---> ...
Best regards,
JSG.
__________