Lothar Esser - cp's OEIS frontend

User: Lothar Esser

Lothar Esser has authored 2 sequences.

A347986 Length of identical sequence of prime / nonprime numbers left and right of the integer n (excluded).

0, 0, 0, 1, 2, 2, 1, 0, 5, 0, 1, 8, 1, 0, 9, 0, 1, 6, 1, 0, 3, 0, 3, 0, 1, 4, 1, 0, 1, 18, 1, 0, 1, 4, 1, 0, 3, 0, 3, 0, 1, 12, 1, 0, 3, 0, 3, 0, 1, 6, 1, 0, 7, 0, 1, 4, 1, 0, 1, 10, 1, 0, 1, 4, 1, 0, 3, 0, 3, 0, 1, 4, 1, 0, 1, 4, 1, 0, 3, 0, 9, 0, 3, 0, 1, 6, 1, 0, 5, 0, 1, 2, 7, 2, 1, 0, 3, 0, 3

Offset: 1

Lothar Esser, Sep 22 2021

nonn

			For n=2 (first useful term) the result is 0 because 2 is preceded by 1 which is by definition nonprime and succeeded by 3 which is prime meaning symmetry is broken right away. A better example may be 5 with a value of 2. The two numbers preceding 5 are 3, 4: prime, nonprime and the succeeding values are 6 and 7 being nonprime and prime. In other words, starting from 5 as center, the first positions are 4 (left) and 6 (right), both nonprimes. The next positions are 3 and 7, both primes. The sequence is now 2 long. It breaks after that because 2 is prime but 8 is nonprime. So we note 2 or 5. Very interesting is 30 which has a sequence of 18 on each side that follow the same pattern.
From _Jon E. Schoenfield_, Sep 22 2021: (Start)
As shown in the illustration below, where P and N denote prime and nonprime, respectively, the distribution of primes and nonprimes around n=21 is symmetrical in the interval [18, 24] = [21-3, 21+3], but not in the interval [17, 25] = [21-4, 21+4] (since 17 is prime but 25 is composite), so a(21) = 3:
.
              |<------- 3 -------->|<------- 3 -------->|
      17     18     19     20     21     22     23     24     25
  -----+------+------+------+------+------+------+------+------+-----
       P      N      P      N      N      N      P      N      N
              |      |      |             |      |      |
              |      |      +-------------+      |      |
              |      +---------------------------+      |
              +-----------------------------------------+
(End)

Cf. A343730, A346399 (number of primes in the interval n +- a(n)).

Table[s={n-1,n+1};k=0;While[SameQ@@PrimeQ@s,k++;s=s+{-1,+1}];k,{n,2,85}] (* Giorgos Kalogeropoulos, Sep 23 2021 *)

f(n) = {my (nb = 0, fL, fR); fL = n\2; if (n%2, fR = fL+1, fL--; fR = fL+2); for (i=0, oo, if (isprime(fL-i) != isprime(fR+i), break, nb++);); if (n%2, 2*nb, 2*nb+1);} \\ A343730
a(n) = (f(2*n) - 1)/2; \\ Michel Marcus, Sep 24 2021

from sympy import *
seq_pole = []
seq_pole.append(-1) #0
seq_pole.append(-1) #1
for i in range(1, 1000):
    d = 1
    #  Check how far the left is identical to the
    # (mirrored) right
    while isprime(i-d) == isprime(i+d):
        d = d + 1
    dmax = d - 1
    seq_pole.append(dmax)
    # i is the center (index) and dmax is the max extent
    # on each side that is the same (or mirrored at i if you will)
    print("{}".format(dmax))

From Jon E. Schoenfield, Sep 22 2021: (Start)
a(n) = Max_{k : A010051(n-j) = A010051(n+j) for 0 <= j <= k}.
a(n) = (A343730(2*n) - 1)/2. (End)

More terms from Michel Marcus, Sep 24 2021

A273802 Prime numbers formed by successively prepending prime numbers to 3.

Original entry on oeis.org

3, 53, 1153, 311153, 101311153, 271101311153, 347271101311153, 631347271101311153, 719631347271101311153, 829719631347271101311153, 1031829719631347271101311153, 11231031829719631347271101311153, 125911231031829719631347271101311153, 1801125911231031829719631347271101311153

Offset: 1

Lothar Esser, Jun 03 2016

The sequence is related to the existing sequence in which primes are appended so that primes result 2, 23, 2311, 231131, ... (see A240563). The current sequence cannot start with the first prime 2 because it could not be extended since any number > 2 and ending in 2 is a nonprime. So this sequence has to start with 3.
One could also consider analogous sequences starting with any prime greater than 3.
The sequence of primes appended at n-th term is 3, 5, 11, 31, 101, 271, 347, 631, 719, 829, 1031, 1123, 1259, 1801, 1907, 2557, 2591, 2851, 2897, 3301, 3467, 3853, 4157, 4789, 6917, 6991, 7127, 7369, 9767, 13879, 15791, 17239, 19541, 22447, 23663, 25309, 25577, 25873, 29873, 33301, 33713, 34543, 36389, 37159, 39821, 40597, 41453, 41479, 43997, ... - Michael De Vlieger, Jun 03 2016

			Start with 3 as the first term.
a(2) = 53, since the next prime after a(1) = 3 is 5; 5 prepended to 3 gives 53, another prime.
a(3) = 1153, since the next prime after that appended to a(2), i.e., 5, is 7, however 7 appended to a(2) = 753 = 3 * 251. The next prime 11, appended to a(2) gives us 1153, which is prime.

Michael S. Branicky, Table of n, a(n) for n = 1..181

Cf. A240563.

a = {3}; Do[p = NextPrime@ a[[n - 1]]; While[! PrimeQ@ FromDigits@ Join[IntegerDigits@ p, Flatten@ Map[IntegerDigits, Reverse@ a]], p = NextPrime@ p]; AppendTo[a, p], {n, 2, 14}]; FoldList[FromDigits@ Join[IntegerDigits@ #2, IntegerDigits@ #1] &, a] (* Michael De Vlieger, Jun 03 2016 *)

from itertools import islice
from sympy import isprime, nextprime
def agen(): # generator of terms
    p = an = 3
    while True:
        yield an
        s = str(an)
        while not isprime(int(str(p) + s)): p = nextprime(p)
        an = int(str(p) + s)
print(list(islice(agen(), 14))) # Michael S. Branicky, Oct 29 2022

#! /usr/bin/tclsh
set prime_list_file list_prime_1000.dat ;
proc PR_read_primes { fh } {
global Prime Nprime;
  set idx 0;
  while { ![eof $fh] } {
    gets $fh line;
    foreach p $line {
      set Prime($idx) $p;
      incr idx;
    }
  }
  set Nprime $idx;
}
proc PR_is_prime { num } {
  set channel [open "| factor $num r"];
  fconfigure $channel -buffering none;
  set line [read $channel] ;
  #puts "$line [llength $line]";
  if { [llength $line] == 2 } {
    catch { close $channel}
    return 1;
  }
  return 0;
}
### main
if { ! [catch "open $prime_list_file r" fh ] } {
  PR_read_primes $fh;
  close $fh;
} else {
    puts "Cannot open file $prime_list_file";
    exit 1
}
set t $Prime(1);
set num_tested_primes 0;
for { set idx 2 } { $idx < 1000 } { incr idx } {
  # Assemble
  # Simple tests
  set s $Prime($idx)$t;
  if { [PR_is_prime $s] } {
    set t $s;
    puts "$t prepended prime $Prime($idx) skipped $num_tested_primes";
    set num_tested_primes 0;
  } else {
     incr num_tested_primes;
  }
}
# The language is Tcl but it requires and external file with the first 1000 primes for convenience. It also uses UNIX program factor as external function to find out whether the number is a prime.

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User: Lothar Esser

A347986 Length of identical sequence of prime / nonprime numbers left and right of the integer n (excluded).

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