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,

….

- - - - - - - - - - - -

(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,...

 

(Iíve also dropped the reference to "filling" squares - weíre ultimately just counting digits here. Filling squares is the idea that got you here, but not the essence of what youíve gotten to. You should mention it in a comment.)

 

Donít forget to cross-reference A058183.

 

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[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 Ericís 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:

 

Ė 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

 

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[Ed Jeffery]:

 

NJAS>I think we should use both styles for both sequences, so we will have four new sequences in all.

 

Also, Ericís 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

 

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[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]:

 

Shouldnít it be2^6*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 Ericís 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.

 

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