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User: Jason Rodgers

Jason Rodgers has authored 3 sequences.

A354440 Digitally delicate primes where the number of digits appended on the left needed to get a prime increases.

294001, 604171, 971767, 2690201, 10564877, 104097043, 354975121, 1378229029, 1444623667, 1594371379, 3979115747, 15737262803, 22090236251, 28198307351, 35373071549, 49430022721, 67580736437, 142243533671, 659956292591, 1385321944133

Offset: 1

Jason Rodgers, May 29 2022

Digitally delicate primes (A050249) are primes such that if any single digit is changed the new number is composite. This sequence gives the smallest such prime that needs more digits added to the left to get to another prime. While this list is not complete it has been shown to be finite.  A widely digitally delicate prime is known which never becomes prime regardless of the number of extra digits.
         294001 can add 1
         604171 can add 3
         971767 can add 4
        2690201 can add 5
       10564877 can add 6
      104097043 can add 7
      354975121 can add 10
     1378229029 can add 11
     1444623667 can add 12
     1594371379 can add 14
     3979115747 can add 15
    15737262803 can add 16
    22090236251 can add 20
    28198307351 can add 26
    35373071549 can add 27
    49430022721 can add 28
    67580736437 can add 30
   142243533671 can add 47
   659956292591 can add 59
  1385321944133 can add 76

			You can add any 1 extra digit on the left to 294001 without getting a prime but adding two digits would allow for the creation of a prime. For example 10294001 is prime but none of X294001 are.
Starting at 604171 you could add 3 extra digits to the left but not 4 without being able to produce a prime number.
X604171 is not prime
X0604171 is not prime
X00604171 is not prime
however 4000604171 is a prime number
For the largest one found so far
X1385321944133
X01385321944133
X001385321944133
...
X000000000000000000000000000000000000000000000000000000000000000000000000001385321944133 are all composite
but 900000000000000000000000000000000000000000000000000000000000000000000000000001385321944133 is prime

Michael Filaseta and Jeremiah Southwick, Primes that become composite after changing an arbitrary digit, Math. Comp. (2021) Vol. 90, 979-993. doi:10.1090/mcom/3593

Cf. A050249 (digitally delicate primes).

import java.math.BigInteger;
public class delicateprimes {
public static void main(String[] args) {
  BigInteger i,reci=new BigInteger("0");
  i= new BigInteger("1");
  long count=0, v,rec=-1;
  for(long loop=1;;loop++)
  {
    i=i.nextProbablePrime();
    v = delicate(i,true);
    if(v>rec) {count++; rec=v;reci=i;System.out.println("REC=("+reci+", "+rec+") " +loop +"   "+count);}
    if(loop%100000==0)System.out.println("Still running, last prime seen was "+i);
  }
}
static int delicate(BigInteger a,boolean f) // Returns how many digits can be tacked on the delicate prime. f=false just tests the prime with no extra 0s
{
  int e, length,max=200;
  if(!f)max=1;
  String num="", num2="";
  if(!prime(a))return -1;
  for(e=0;e0)length=e;else length = num.length();
    for(int j=0;j

Partially edited by N. J. A. Sloane, Sep 03 2022

A129913 a(n) is the number of ways of producing n, with an expression like: +- 1 # 2 # 3 # 4 # 5 # 6 # 7 # 8 # 9, where each # is replaced by any one of +,-,*,/; and the value is computed going left to right.

Original entry on oeis.org

694, 299, 143, 171, 156, 156, 183, 155, 167, 434, 160, 140, 129, 111, 89, 147, 114, 201, 171, 94, 87, 90, 68, 95, 130, 130, 44, 120, 41, 72, 76, 72, 50, 81, 50, 53, 137, 49, 46, 78, 41, 75, 36, 61, 28, 123, 35, 86, 47, 40, 40, 47, 40, 37, 109, 61, 32, 58, 23, 53

Offset: 0

Jason Rodgers (Jason628(AT)gmail.com), Jun 05 2007, Jun 06 2007

			Going left-to-right, +1-2/3-4-5*6/7+8*9 = 0; and there are 694 such expressions which yield zero, so a(0)=694.

Don Reble, Table of n, a(n) for n = 0..1000
Don Reble, C++ program to compute sequence

Corrected and edited by Don Reble, Jun 23 2008

A090230 Decimal positions where Pi, E and Phi are the same.

Original entry on oeis.org

12, 99, 169, 395, 499, 595, 606, 693, 824, 827, 840, 940, 1282, 1291, 1384, 1594, 1705, 1742, 1905, 2020, 2060, 2153, 2257, 2302, 2359, 2367, 2507, 2546, 2557, 2710, 2724, 2791, 2832, 2857, 3036, 3051, 3280, 3309, 3429, 3497, 3518, 3591, 3651, 3709, 3867

Offset: 0

Jason Rodgers (Jason628(AT)hotmail.com), Jan 22 2004

Harvey P. Dale, Table of n, a(n) for n = 0..1000

dpQ[{a_,b_,c_}]:=a==b==c; Module[{nn=4000,p,e,f},p=RealDigits[Pi,10,nn][[1]]; e=RealDigits[E,10,nn][[1]];f=RealDigits[GoldenRatio,10,nn][[1]]; Flatten[Position[Thread[{p,e,f}],?dpQ]]-1] (* _Harvey P. Dale, Dec 20 2012 *)

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User: Jason Rodgers

A354440 Digitally delicate primes where the number of digits appended on the left needed to get a prime increases.

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A129913 a(n) is the number of ways of producing n, with an expression like: +- 1 # 2 # 3 # 4 # 5 # 6 # 7 # 8 # 9, where each # is replaced by any one of +,-,*,/; and the value is computed going left to right.

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A090230 Decimal positions where Pi, E and Phi are the same.

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Mathematica