A228325 -id:A228325

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A068695

Smallest number (not beginning with 0) that yields a prime when placed on the right of n.

+10
11

1, 3, 1, 1, 3, 1, 1, 3, 7, 1, 3, 7, 1, 9, 1, 3, 3, 1, 1, 11, 1, 3, 3, 1, 1, 3, 1, 1, 3, 7, 1, 17, 1, 7, 3, 7, 3, 3, 7, 1, 9, 1, 1, 3, 7, 1, 9, 7, 1, 3, 13, 1, 23, 1, 7, 3, 1, 7, 3, 1, 3, 11, 1, 1, 3, 1, 3, 3, 1, 1, 9, 7, 3, 3, 1, 1, 3, 7, 7, 9, 1, 1, 9, 19, 3, 3, 7, 1, 23, 7, 1, 9, 7, 1, 3, 7, 1, 3, 1, 9, 3, 1

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

Max Alekseyev (see link) shows that a(n) always exists. Note that although his argument makes use of some potentially large constants (see the comments in A060199), the proof shows that a(n) exists for all n. - N. J. A. Sloane, Nov 13 2020

Many numbers become prime by appending a one-digit odd number. Some numbers (such as 20, 32, 51, etc.) require a 2-digit odd number (A032352 has these). In the first 100000 values of n there are only 22 that require a 3-digit odd number (A091089). There probably are some values that require odd numbers of 4 or more digits, but these are likely to be very large. - Chuck Seggelin (barkeep(AT)plastereddragon.com), Dec 18 2003

LINKS

David W. Wilson, Table of n, a(n) for n=1..10000

Max Alekseyev, Given n, there is a k such that the concatenation n||k is a prime, Nov 09 2020

Index entries for primes involving decimal expansion of n

EXAMPLE

a(20)=11 because 11 is the minimum odd number which when appended to 20 forms a prime (201, 203, 205, 207, 209 are all nonprime, 2011 is prime).

MATHEMATICA

d[n_]:=IntegerDigits[n]; t={}; Do[k=1; While[!PrimeQ[FromDigits[Join[d[n], d[k]]]], k++]; AppendTo[t, k], {n, 102}]; t (* Jayanta Basu, May 21 2013 *)

mon[n_]:=Module[{k=1}, While[!PrimeQ[n*10^IntegerLength[k]+k], k+=2]; k]; Array[mon, 110] (* Harvey P. Dale, Aug 13 2018 *)

PROG

(PARI) A068695=n->for(i=1, 9e9, ispseudoprime(eval(Str(n, i)))&&return(i)) \\ M. F. Hasler, Oct 29 2013

(Python)

from sympy import isprime

from itertools import count

def a(n): return next(k for k in count(1) if isprime(int(str(n)+str(k))))

print([a(n) for n in range(1, 103)]) # Michael S. Branicky, Oct 18 2022

CROSSREFS

Cf. A032352 (a(n) requires at least a 2 digit odd number), A091089 (a(n) requires at least a 3 digit odd number).

Cf. also A060199, A228325, A336893.

KEYWORD

base,easy,nonn

AUTHOR

Amarnath Murthy, Mar 03 2002

EXTENSIONS

More terms from Chuck Seggelin (barkeep(AT)plastereddragon.com), Dec 18 2003

Entry revised by N. J. A. Sloane, Feb 20 2006

More terms from David Wasserman, Feb 14 2006

STATUS

approved

A228323

a(1)=1; thereafter a(n) is the smallest number m not yet in the sequence such that at least one of the concatenations a(n-1)||m or m||a(n-1) is prime.

+10
6

1, 3, 2, 9, 5, 21, 4, 7, 6, 13, 10, 19, 16, 27, 8, 11, 15, 23, 12, 17, 20, 29, 14, 33, 26, 47, 18, 31, 25, 39, 22, 37, 24, 41, 30, 49, 34, 57, 28, 43, 36, 59, 32, 51, 38, 53, 42, 61, 45, 67, 58, 69, 55, 63, 44, 81, 35, 71, 48, 77, 50, 87, 62, 99, 40, 73, 46

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

Does every number appear in the sequence?

If a(n) is coprime to 10, then a(n+1) exists by Dirichlet's theorem. - Eric M. Schmidt, Aug 20 2013 [In more detail: let a(n) have d digits, and consider the arithmetic progression k*10^d + a(n), and apply Dirichlet's theorem. This gives a number k such that the concatenation k||a(n) is prime. N. J. A. Sloane, Nov 08 2020]

The argument in A068695 shows that a(n) always exists. - N. J. A. Sloane, Nov 11 2020

LINKS

Alois P. Heinz, Table of n, a(n) for n = 1..10000

Eric Angelini, Primes by concatenation, Posting to the Sequence Fans Mailing List, Aug 14 2013.

Michael De Vlieger, Labeled log-log scatterplot of a(n) n = 1..2^14, showing m coprime to 10 in red, otherwise dark blue.

Index entries for primes involving decimal expansion of n

MATHEMATICA

f[s_] := Block[{k = 2, idj = IntegerDigits@ s[[-1]]}, While[idk = IntegerDigits@ k; MemberQ[s, k] || ( !PrimeQ@ FromDigits@ Join[idj, idk] && !PrimeQ@ FromDigits@ Join[idk, idj]), k++]; Append[s, k]]; Nest[f, {1}, 66] (* Robert G. Wilson v, Aug 20 2013 *)

PROG

(Python)

from sympy import isprime

from itertools import islice

def c(s, t): return isprime(int(s+t)) or isprime(int(t+s))

def agen():

aset, k, mink = set(), 1, 2

while True:

an = k; aset.add(an); yield an; s, k = str(an), mink

while k in aset or not c(s, str(k)): k += 1

while mink in aset: mink += 1

print(list(islice(agen(), 56))) # Michael S. Branicky, Oct 17 2022

CROSSREFS

See A228324 for the primes that arise.

Cf. A069695, A228325.

KEYWORD

nonn,base

AUTHOR

N. J. A. Sloane, Aug 20 2013

EXTENSIONS

More terms from Alois P. Heinz, Aug 20 2013

STATUS

approved

A338366

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

COMMENTS

See A090287 for more information.

From Robert Price, Sep 20 2023: (Start)

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)

LINKS

Robert Price, Table of n, a(n) for n = 0..365

Index entries for primes involving decimal expansion of n

EXAMPLE

a(3) = 1 because 131 is prime.

a(4) = 11 because 11411 is prime, and all of 141, 242, 343, ..., 949 are composite.

CROSSREFS

Cf. A090287.

Related sequences: A010785, A068695, A091088, A228323, A228325, A336893, A338712 (see also the Index link above).

KEYWORD

sign,base

AUTHOR

N. J. A. Sloane, Nov 08 2020

EXTENSIONS

More terms from Alois P. Heinz, Nov 08 2020

STATUS

approved

A245727

Least number k >= 0 such that n concatenated with n + k is prime.

+10
1

0, 1, 4, 3, 4, 1, 2, 1, 2, 3, 6, 1, 6, 9, 8, 3, 4, 5, 12, 7, 8, 15, 10, 13, 6, 7, 2, 5, 10, 7, 6, 19, 10, 15, 4, 1, 2, 9, 4, 9, 12, 1, 6, 3, 2, 3, 4, 13, 2, 1, 2, 9, 28, 17, 2, 1, 22, 3, 22, 7, 2, 1, 4, 5, 4, 7, 12, 1, 2, 9, 6, 11, 20, 3, 2, 5, 12, 1, 14, 1, 10, 5, 4, 37, 12, 3, 16, 5, 10

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,3

LINKS

Alois P. Heinz, Table of n, a(n) for n = 1..10000

FORMULA

a(n) = A228325(n) - n for n > 1.

EXAMPLE

33 is not prime. 34 is not prime. 35 is not prime. 36 is not prime. 37 is prime. Since 7 is 4 more than 3, a(3) = 4.

MAPLE

a:= proc(n) local j; for j from n do if isprime(n*10^(1+ilog10(j))+j) then return(j-n) fi od end proc:

seq(a(n), n=1..100); # Robert Israel, Jul 30 2014

MATHEMATICA

lnk[n_]:=Module[{k=0, idn=IntegerDigits[n]}, While[!PrimeQ[FromDigits[ Join[ idn, IntegerDigits[ n+k]]]], k++]; k]; Array[lnk, 90] (* Harvey P. Dale, Oct 05 2014 *)

PROG

(PARI)

a(n) = for(k=n, 10^4, if(isprime(eval(concat(Str(n), Str(k)))), return(k-n)))

vector(150, n, a(n))

(Python)

def a(n):

..for k in range(n, 10**4):

....if isprime(str(n)+str(k)):

......return k-n

n = 1

while n < 150:

..print(a(n), end=', ')

..n += 1

CROSSREFS

Cf. A228325.

KEYWORD

nonn,base

AUTHOR

Derek Orr, Jul 30 2014

STATUS

approved

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