| Displaying 1-7 of 7 results found.
page
1
Smallest nonnegative number k not starting or ending with the digit 1 that forms a prime when it is sandwiched between n ones to the left of k and n ones to the right of k.
+10
4
0, 3, 4, 8, 36, 8, 5, 72, 28, 6, 79, 212, 23, 6, 73, 24, 52, 62, 3, 28, 220, 53, 75, 58, 228, 9, 265, 89, 214, 86, 215, 4, 7, 39, 295, 40, 87, 216, 97, 6, 264, 53, 287, 223, 4, 239, 259, 25, 57, 364, 49, 38, 93, 86, 27, 30, 80, 24, 6, 356, 50, 645, 395, 206
COMMENTS
n = 1 is the only case where a(n) = 0, since for any n > 1, A138148(n) is divisible by A002275(n). If k has an even number of digits and is a multiple of 11, then k is not a term. If k = (10^r+1)(10^m-1)/9 for some m > 0, r >= 0, then k is not a term. If A272232(k) = 0, then k is not a term. - Chai Wah Wu, Nov 08 2019
EXAMPLE
a(1) = 0, because 101 is prime.
a(5) = 36, because the smallest x >= 0 such that 11111_x_11111 (where '_' denotes concatenation) is prime is 36. The decimal expansion of that prime is 111113611111.
MATHEMATICA
Table[k = 0; s = Table[1, {n}]; While[Or[!PrimeQ[FromDigits[s ~Join~ IntegerDigits[k] ~Join~ s]], Or[First@ IntegerDigits@ k == 1, Last@ IntegerDigits@ k == 1]], k++]; k, {n, 64}] (* Michael De Vlieger, May 28 2015 *)
PROG
(PARI) a000042(n) = (10^n-1)/9
a(n) = my(k=0); while(k==10 || k%10==1 || k\(10^(#Str(k)-1))==1 || !ispseudoprime(eval(Str(a000042(n), k, a000042(n)))), k++); k
Smallest prime obtained by appending a number with identical digits to n or 0 if no such prime exists.
+10
3
2, 11, 23, 31, 41, 53, 61, 71, 83, 97, 101, 113, 127, 131, 149, 151, 163, 173, 181, 191, 2011, 211, 223, 233, 241, 251, 263, 271, 281, 293, 307, 311, 3299, 331, 347, 353, 367, 373, 383, 397, 401, 419, 421, 431, 443, 457, 461, 479, 487, 491, 503, 511111, 521, 5333
COMMENTS
For n <= 15392, a(n) = 0 if and only if n = 6930. Conjecture: if a(n) = 0, then n is divisible by 3. Conjecture verified for n <= 10^6. a(n) = 0 for n = 6930, 50358, 56574, 72975.
PROG
(Python)
from gmpy2 import mpz, digits, is_prime
....if n in (6930, 50358, 56574, 72975):
........return 0
....if n == 0:
........return 2
....sn = str(n)
....for i in range(1, limit+1):
........for j in range(1, 10, 2):
............si = digits(j, 10)*i
............p = mpz(sn+si)
............if is_prime(p):
................return int(p)
....else:
........return 'search limit reached.'
Numbers with all digits equal and from the set {1, 3, 7, 9}.
+10
3
1, 3, 7, 9, 11, 33, 77, 99, 111, 333, 777, 999, 1111, 3333, 7777, 9999, 11111, 33333, 77777, 99999, 111111, 333333, 777777, 999999, 1111111, 3333333, 7777777, 9999999, 11111111, 33333333, 77777777, 99999999, 111111111, 333333333, 777777777, 999999999, 1111111111, 3333333333, 7777777777, 9999999999
COMMENTS
Candidates for prefixes and suffixes in A090287.
FORMULA
G.f.: x*(1 + 3*x + 7*x^2 + 9*x^3) / ((1 - x)*(1 + x)*(1 + x^2)*(1 - 10*x^4)).
a(n) = 11*a(n-4) - 10*a(n-8) for n>8.
(End)
MAPLE
a:= n-> [1, 3, 7, 9][1+irem(n-1, 4)]*(10^iquo(n+3, 4)-1)/9:
Smallest palindromic prime by adding at least one digit to both the left and right of n
+10
2
101, 313, 727, 131, 11411, 151, 10601, 373, 181, 191, 30103, 1114111, 1120211, 11311, 11411, 31513, 1160611, 1117111, 18181, 71917, 30203, 1120211, 72227, 32323, 12421, 1250521, 36263, 12721, 12821, 39293, 10301, 11311, 32323, 13331, 14341, 33533, 16361, 77377
PROG
(Python)
from __future__ import division
from sympy import isprime
def palgenrange2(m, l): # generator of odd-length palindromes of length at least m and at most 2*l
....if m == 1:
........yield 0
....for x in range(m//2+1, l+1):
........n = 10**(x-1)
........for y in range(n, n*10):
............s = str(y)
............yield int(s+s[-2::-1])
....sn = str(n)
....for p in palgenrange2(len(sn)+2, len(sn)+20):
........if sn in str(p)[1:-1] and isprime(p):
............break
....else:
........return 'search limit reached'
EXTENSIONS
a(10), a(12), a(16), a(18) corrected by Chai Wah Wu, Mar 22 2015
Smallest prime obtained by appending n to a nonzero number with identical digits or 0 if no such prime exists.
+10
2
0, 11, 0, 13, 0, 0, 0, 17, 0, 19, 0, 211, 0, 113, 0, 0, 0, 317, 0, 419, 0, 421, 0, 223, 0, 0, 0, 127, 0, 229, 0, 131, 0, 233, 0, 0, 0, 137, 0, 139, 0, 241, 0, 443, 0, 0, 0, 347, 0, 149, 0, 151, 0, 353, 0, 0, 0, 157, 0, 359, 0, 461, 0, 163, 0, 0, 0, 167, 0, 269
COMMENTS
a(n) = 0 if n is even or a multiple of 5. Conjecture: all other terms are nonzero. Conjecture verified for n <= 10^7.
"Appending" means "on the right".
PROG
(Python)
from gmpy2 import digits, mpz, is_prime
....sn = str(n)
....if not (n % 2 and n % 5):
........return 0
....for i in range(1, limit+1):
........for j in range(1, 10):
............si = digits(j, 10)*i
............p = mpz(si+sn)
............if is_prime(p):
................return int(p)
....else:
........return 'search limit reached.'
a(n) = smallest positive number k with all digits equal such that the concatenation k||n||k is prime, or -1 if no such k exists.
+10
2
1, 3, 7, 1, 11, 1, 7777, 3, 1, 1, 9, -1, 7, 33, 99, 1, 3, 1, 1, 9, 1, 11, -1, 1, 7, 3, 7777777777, 1111, 111, 1, 1, 3, 1, -1, 3, 33, 1, 3, 1, 77777777777777, 111, 3, 1111111111111111111111111111111111111111, 3, -1, 1, 3, 1, 1, 999, 7, 1, 11, 1, 7, -1, 33, 1, 3, 3, 1, 3, 1
COMMENTS
For a(366), k is a string of 8441 1's.
The sequence then continues: 77, 1, 1, 3, 1, 1, 9, 7777777, 1, 11, 3, 1, 11, 9, 77, 11111, 1, 1, 33333, 3, 7, 9, 3, 1, 77, 1, 1, 9, 7777777777 until a(396) where k is a sequence of 269 1's.
The sequence then continues: 9, 777, 11, 9, 1, 7, 3, 7, 1, 11, 1, 1, 9, 9, 1111, 3, 999, 77777, 99, 7, 7, 3, 7, -1, 3, 1, 11, 77, 1, 77, 3, 1, 7, 3, 3, 1, 111111, 1, 7, 99, 7, 1111, 9, 1, 1, 11, 1, 7777777, 11, 1, 1111, 3, 1111, 7, 3, 7, 11, 3, 1, 1, 111, 3, 1, 3, 3, 1, 33, 9, 11, 33, 3, 7, 3, 3, 7, 99, 1, 1, 11, 3, 1, 9, 7, 77, 9, 1, 1, 3, 1, 7777, 33, 3, 1, 33, 3, 77, 77, 9, 1, 3, 33, 11111, 9, 9. (End)
EXAMPLE
a(3) = 1 because 131 is prime.
a(4) = 11 because 11411 is prime, and all of 141, 242, 343, ..., 949 are composite.
Smallest prime of the form "Concatenate(m,n,m)".
+10
1
101, 313, 727, 131, 11411, 151, 13613, 373, 181, 191, 9109, 131113, 7127, 171317, 131413, 1151, 3163, 1171, 1181, 9199, 1201, 112111, 172217, 1231, 7247, 3253, 372637, 172717, 232823, 1291, 1301, 3313, 1321, 233323, 3343, 273527, 1361, 3373, 1381, 173917, 174017
EXAMPLE
111 is divisible by 3, and 212 is divisible by 2, but 313 is prime; therefore, a(1) = 313.
MAPLE
f:= proc(n) local dn, x, dx, p;
dn:= 10^(1+ilog10(n));
for x from 1 by 2 do if igcd(x, n) = 1 then
dx:= 10^(1+ilog10(x));
p:= x*(1+dx*dn)+n*dx;
if isprime(p) then return(p) fi
fi od
end proc:
# second Maple program:
a:= proc(n) local m, p; for m do
p:= parse(cat(m, n, m));
if isprime(p) then break fi od; p
end:
MATHEMATICA
mnmPrimes = {}; f[m_, n_] := FromDigits[Flatten[{IntegerDigits[m], IntegerDigits[n], IntegerDigits[m]}]]; Do[m = 1; While[True, If[PrimeQ[f[m, n]], AppendTo[mnmPrimes, f[m, n]]; Break[]]; m+=2], {n, 0, 40}]; mnmPrimes
PROG
(PARI) a(n) = {m=1; while (! isprime(p=eval(concat(Str(m), concat(Str(n), Str(m))))), m+=2); p; } \\ Michel Marcus, Mar 23 2015 (Sage)
m = 1
sn = str(n)
while True:
sm = str(m)
a = int(sm + sn + sm)
if is_prime(a):
return a
m += 2
(Haskell)
a252942 n = head [y | m <- [1..],
let y = read (show m ++ show n ++ show m) :: Integer, a010051' y == 1]
Search completed in 0.014 seconds
|