[seqfan] Re: Squares = concatenation of primes.

[f.firoozbakht at sci.ui.ac.ir]

  Dear Zak and Neil,

  Finding N(k,k,1) isn't hard. Because we can easily prove that square numbers
  greater than 25 such that all their digits are primes are of the form
  100*m^2+100*m+25 where mod(m,10) is one of the numbers 1, 3, 6 & 8 (I don't
  mean that digits of all such numbers are primes).

  Let S(k) be the set of all k-digit square numbers such that all their
  digits are primes. So according to the definition N(k,k,1)=length(S(k)).

  Also if k is an odd number greater than 1 then we can easily show that
  (25*10^(k-1)-5*10^((k+1)/2)+25)/9 is in the set S(k).

  I think the sequence a(k)=N(k,k,1) is interesting and we can submit it.
  Do you agree?

  a(k)=N(k,k,1) for k=1,2,,...,18 are:

  0, 1, 1, 1, 2, 1, 2, 3, 8, 6, 8, 13, 13, 15, 15, 13, 19, 34

  {k, N(k,k,1), S(k)} for k=1,2,3,...,18 are:

  {1, 0, {}}

  {2, 1, {25}}

  {3, 1, {225}}

  {4, 1, {7225}}

  {5, 2, {27225,55225}}

  {6, 1, {235225}}

  {7, 2, {2772225,3553225}}

  {8, 3, {23377225,33235225,57532225}}

  {9, 8, {227557225,252333225,277722225,337273225,357777225,
   523723225,735223225,777573225}}

  {10, 6, {2523555225,3325252225,3377353225,5232352225,7333353225,7723773225}}

  {11, 8, {22775337225,27777222225,33275232225,37727235225,37777753225,
   52732233225,57775333225,72272257225}}

  {12, 13, {253527355225,257723752225,275535757225,323573257225,325373272225,
   333523575225,337752757225,523575252225,725333272225,727327537225,
   732333735225,757535233225,773757733225}}

  {13, 13, {2277337537225, 2527352755225, 2527575327225, 2577357322225,
   2723275555225,2777772222225,3235555525225,3535772533225,3572723727225,
   5227327732225,5335337727225,5523275527225,5557735525225}}

  {14, 15, {22225237353225,22535575537225,22732727373225,25225355575225,
   25372527523225,25772527522225,27252732772225,32722575733225,
   32773735777225,33233322577225,35773377777225,52723355377225,
   52725533725225,77535333322225,77573553532225}}

  {15, 15, {222532253775225,223273375252225,227722737535225,232555332555225,
   235732572352225,255757337532225,277777722222225,327737255355225,
   332375377257225,333753255277225,335333555773225,353725253532225,
   377327322757225,575237772333225,772775227357225}}

  {16, 13, {2227522355323225,2233775572237225,2277233235733225,
   2357732235555225,2557232357737225,2577337533225225,2775552752755225,
   3227357352727225,3753753732373225,5535533353357225,7337235573252225,
   7773355527223225,7773757572277225}}

  {17,19,{22752755377227225,23377527775323225,23525757235777225,
   23557373557333225,23725522775737225,23733277325527225,25222723552723225,
   25255275577237225,27232552573723225,27522257555772225,27537532275253225,
   27727273532557225,27737375257522225,27777777222222225,35232553232323225,
   52573337755225225,55222727573377225,73722222532273225,77525222275255225}}

  {18, 34, {227353273752535225,227532353322355225,227735573555755225,
   232775723225773225,255527333273323225,257332577357772225,257555757725235225,
   257572373527333225,257727727257277225,272523553375273225,272555377733773225,
   273525323322355225,273555323223277225,273732573773322225,275572255773555225,
   277553552527255225,277575753735522225,322235735525353225,327575233325353225,
   327752225337355225,327773522552557225,337737257732272225,352333257232522225,
   352773733727332225,357233533337727225,537537537725335225,553353757322575225,
   555235355777557225,557755773325335225,572533727572372225,722555225555275225,
   733372377232573225,775237572752335225,777773335235722225}}


  Best wishes,
  Farideh


Quoting zak seidov <zakseidov at yahoo.com>:

> There are 15 squares that are concatenations of two 3-digit primes:
>
> 101761, 113569, 127449, 131769, 137641, 149769, 167281, 199809,   
> 349281, 439569, 463761, 491401, 641601, 683929, 797449
>
> What about other cases of s-digit squares that are concatenations of  
>  n primes with d digits (s=n*d)?
>
> In our case s=6, n=2, d=3 there we have N(s,n,d) = 15.
>
> Also:
> N(2,2,1)=1 (one square 25 with prime digits)
> N(3,3,1)=1 (one square 225 with prime digits )
> N(4,4,1)=1 (one square 7225 with prime digits)
> N(4,2,2)=3 (three squares 4761, 5329, 5929 with primes  47,61,    
> 53,29,  59,29}
> N(5,5,1)=2  (two squares 27225, 55225 with all digits prime}
> N(6,6,1)=1  ( one square 235225 with prime digits)
> N(6,3,2)=6 {six squares 136161, 178929, 231361, 534361, 591361, 677329
> primes 13,61,61,  17,89,29,  23,13,61,  53,43,61,  59,13,61,  67,73,29}
> Notice that we don't count square 670761 as 07 isn't 2-digit prime.
> .........
> N(9,3,3)=68 (68 squares all concatenations of three 3-digit primes:
> 103449241, 107557641, 109139809, 109181601, 109223401, 127757809,    
> 139263601, 139877929, 151757761, 157577809, 179211769, 191241241,   
> 191739409, 193293409, 193571569, 199967881, 211673401, 227617569,   
> 227919409, 229613409, 251127409, 269977761, 283619281, 283821409,   
> 307967401, 337199769, 337787641, 347337769, 349577809, 349727401,    
> 373223761, 373919569, 383337241, 449397601, 467467641, 491641929,   
> 503149761, 509359761, 509811241, 521163241, 523997881, 569251881,   
> 571353409, 593263449, 607277449, 617373409, 619661449, 643281769,   
> 653773761, 709103641, 733163929, 751911241, 761263281, 877877641,   
> 881911809, 919241761, 967023409, 967769881, 971631241, 997233241,   
> 997991281)
>
> Anyone may wish to check/extend/submit these to OEIS?
> thx, zak
>
>
>
>
>
>
>
> _______________________________________________
>
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>









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