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[ 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 |
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Sequence [k] in order of
increasing prepended digits 1's.
Each value < n in [d^^n][k]
must be composite.
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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 |
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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] | |
| | | |
| | | |
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[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 |
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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.
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Sequence [k] in order of
increasing prepended digits 9's.
Each value < n in [d^^n][k]
must be composite.
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\\ 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;
)
}
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