[ May 20, 2023 ] [ Last update June 11, 2023 ]
Prepending n digits d to k so that [d^^n][k] becomes
a record delayed prime
Xinyao Chen (email)

While WONplate 197 is appending digits to the RIGHT of a number,
now let us consider prepending digits to the LEFT of the number.
Except for 0, all digits 1 to 9 can be used, but of course we only
consider the numbers k both coprime to 10 and coprime to d (since all
numbers [d^^n][k] are divisible by gcd(k,10) and gcd(k,d)).

Here are the record k-values for each case in base 10.

( '^^' is symbol for concatenation )

Also when numbers get larger I will accept PRP (PRobable Prime)
as valid entries.

Click on the header titles to open the related worksheets from Xinyao Chen.
The main page is at → https://sites.google.com/view/world-of-numbers-plate-219/

BASE 10
Prime by prepending 1's	Prime by prepending 2's	Prime by prepending 3's
[1^^1][1]	[2^^3][1]	[3^^1][1]
[1^^17][11]	[2^^204][17]	[3^^2][19]
[1^^713][33]	[2^^19151][99]	[3^^4][41]
[1^^3372][1033]	[2^^30843][737]	[3^^7][127]
[1^^35793][2233]	[2^^34697][3717]	[3^^8][157]
[1^^ >100000][2661]	[2^^ >100000][3737]	[3^^34][443]
		[3^^24037][781]
		[3^^ >100000][4037]
[1] to [1033] by Xinyao Chen [2233] & [2661] by PDG	[1] to [99] by Xinyao Chen [737] to [3737] by PDG	[1] to [443] by Xinyao Chen [781] to [4037] by PDG
See also A200065
Sequence [k] in order of increasing prepended digits 1's. Each value < n in [d^^n][k] must be composite.	Sequence [k] in order of increasing prepended digits 2's. Each value < n in [d^^n][k] must be composite.	Sequence [k] in order of increasing prepended digits 3's. Each value < n in [d^^n][k] must be composite.
Prime by prepending 4's	Prime by prepending 5's	Prime by prepending 6's
[4^^1][1]	[5^^11][1]	[6^^1][1]
[4^^2][9]	[5^^69][13]	[6^^5037][11]
[4^^15][11]	[5^^182][61]	[6^^ >100000][1859]
[4^^50][27]	[5^^240][181]
[4^^187][141]	[5^^645][241]
[4^^837][319]	[5^^1683][319]
[4^^2846][609]	[5^^79773][803]
[4^^3112][1261]	[5^^ >100000][1359]
[4^^9688][1431]
[4^^ >100000][1551]
[1] to [1261] by Xinyao Chen [1431] & 1551 by PDG	[1] to [319] by Xinyao Chen [803] & [1359] by PDG	[1] to [11] by Xinyao Chen [1859] by PDG
Sequence [k] in order of increasing prepended digits 4's. Each value < n in [d^^n][k] must be composite.	Sequence [k] in order of increasing prepended digits 5's. Each value < n in [d^^n][k] must be composite.	Sequence [k] in order of increasing prepended digits 6's. Each value < n in [d^^n][k] must be composite.
Prime by prepending 7's	Prime by prepending 8's	Prime by prepending 9's
[7^^1][1]	[8^^2][1]	[9^^2][1]
[7^^3][11]	[8^^3][13]	[9^^1798][13]
[7^^18][29]	[8^^77][33]	[9^^3527][1177]
[7^^64][79]	[8^^96][149]	[9^^8654][2587]
[7^^173][123]	[8^^11592][151]	[9^^ >100000][2873]
[7^^2502][293]	[8^^68805][473]
[7^^21186][821]	[8^^89100][503]
[7^^ >100000][909]	[8^^ >100000][679]
[1] to [293] by Xinyao Chen [821] & [909] by PDG	[1] to [149] by Xinyao Chen [151] to [679] by PDG	[1] to [1177] by Xinyao Chen [2587] & [2873] by PDG
Sequence [k] in order of increasing prepended digits 7's. Each value < n in [d^^n][k] must be composite.	Sequence [k] in order of increasing prepended digits 8's. Each value < n in [d^^n][k] must be composite.	Sequence [k] in order of increasing prepended digits 9's. Each value < n in [d^^n][k] must be composite.

\\ This is my PARI/GP program code for WONplate 224 by PDG [June 2, 2023]
\\ ie. making sequence [k] in order of increasing prepended digits d's.
\\ Each value < n in [d^^n][k] must be composite.
{
d=7;
for(a=1, 100,
     b=1;
     until(flag==0,
          f=(d*(10^a-1)/9)*10^(#(digits(b)))+b;
          while(ispseudoprime(f)==0,
               b+=2; f=(d*(10^a-1)/9)*10^(#(digits(b)))+b;
          );
          \\ print(f); print(a,"  ",b);
          ad=0; flag=0;
          for(i=1, a-1,
               ad++; fd=(d*(10^ad-1)/9)*10^(#(digits(b)))+b;
               if(ispseudoprime(fd), flag++; break(); );
          );
          b+=2;
     );
     print("[",d,"^^",a,"][",b-2,"]");
     b=1;
)
}