A174031 -id:A174031 - cp's OEIS frontend

A174414 a(n) is the smallest natural number k such that the concatenation (n+k)||k is a prime number.

3, 1, 1, 3, 1, 1, 3, 3, 1, 9, 19, 1, 3, 1, 7, 3, 1, 1, 3, 1, 13, 17, 1, 1, 3, 1, 1, 3, 7, 1, 9, 1, 23, 3, 3, 19, 17, 7, 1, 3, 1, 1, 3, 21, 1, 11, 3, 1, 3, 7, 1, 9, 1, 7, 21, 1, 7, 3, 1, 7, 3, 1, 1, 3, 1, 17, 9, 1, 1, 3, 3, 7, 9, 1, 1, 9, 13, 7, 3, 1, 1, 3, 3, 11, 3, 7, 1, 27, 7, 1, 9, 3, 1, 9, 3, 1, 9, 1

Offset: 1

Ulrich Krug (leuchtfeuer37(AT)gmx.de), Mar 19 2010

The last digit of a(n) must be 1, 3, 7 or 9.

If n and 10^d+1 are not coprime, then a(n) cannot have d digits. - Robert Israel, Sep 16 2024

			43 = prime(14) = (3+1)||3, a(1) = 3
31 = prime(11) = (1+2)||1, a(2) = 1
41 = prime(13) = (1+3)||1, a(3) = 1
3413 = prime(480) = (13+21)||13, a(21) = 13
11527 = prime(1390) = (27+88)||27, a(88) = 27
Note cases where consecutive values of n give consecutive primes:
n=17: 181 = prime(42) = (1+17)||1, n=18: 191 = prime(43) = (1+18)||1
k=41: 421 = prime(82) = (1+41)||1, n=42: 431 = prime(83) = (1+42)||1
... are there infinitely many of such?
a(11) = 19, 3019 is a resulting "candidate" for n = 301 - 9 = 292, but a(292) = 3 gives 2953 = prime(425)
First twice resulting prime is 5623 = prime(739) = (23+33)||23 = 5623 = (559+3)||3

Theo Kempermann, Zahlentheoretische Kostproben, Harri Deutsch, 2. aktualisierte Auflage 2005.
Helmut Kracke, Mathe-musische Knobelisken, Dümmler Bonn, 2. Auflage 1983.
Hugo Steinhaus, Studentenfutter, Urania-Verlag Leipzig-Jena-Berlin, 1991.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A089712, A126785, A171154, A173976, A174031.

tcat:= proc(a,b) a*10^(1+ilog10(b))+b end proc:
f:= proc(n) local k, d;
    for d from 1 do
      if igcd(n, 10^d+1) > 1 then next fi;
      for k from 10^(d-1)+`if`(d=1, 0, 1) to 10^d by 2 do
        if isprime(tcat(n+k, k)) then return k fi
    od od
end proc:
map(f, [$1..100]); # Robert Israel, Sep 16 2024

a(n) = my(k=1); while (!isprime(eval(concat(Str(n+k), Str(k)))), k++); k; \\ Michel Marcus, Sep 17 2024

from itertools import count
from math import gcd
from sympy import isprime
def A174414(n):
    for l in count(1):
        if gcd(n,(m:=10**l)+1)==1:
            r = m//10
            a = m*(n+r)+r
            for k in range(r,m):
                if isprime(a):
                    return k
                a += m+1 # Chai Wah Wu, Sep 18 2024

a(n) = 1 for n > 1 in A126785.

Edited and corrected by Robert Israel, Sep 16 2024

A174034 The smallest prime p such that the double-concatenation prime(n) // prime(n+1) // p is a prime number.

Original entry on oeis.org

3, 3, 7, 19, 17, 7, 17, 7, 3, 23, 11, 11, 11, 17, 3, 3, 7, 3, 11, 17, 29, 19, 13, 7, 37, 7, 23, 37, 7, 23, 7, 7, 7, 11, 7, 53, 29, 31, 31, 13, 11, 17, 7, 11, 11, 29, 23, 47, 7, 7, 7, 13, 11, 19, 67, 19, 13, 101, 59, 13, 13, 31, 17, 23, 7, 13, 29, 73, 29, 7

Offset: 1

Eva-Maria Zschorn (e-m.zschorn(AT)zaschendorf.km3.de), Mar 06 2010

It is conjectured that a(n) = 3 for infinitely many n.

			n=1: 2 // 3 // 3 = 233, which is prime, so a(1) = 3.
n=2: 3 // 5 // 2 = 352, which is not prime, but 3 // 5 // 3 = 353 is, so a(2) = 3.

Cf. A030461, A030459, A030469, A171154, A174031

A174034(n)={ n=eval(Str(prime(n),prime(n+1))); for( d=1,99, n*=10; forprime( p=10^(d-1),10^d, isprime(n+p) & return(p)))} \\ M. F. Hasler, Dec 01 2010

concat = lambda xx: Integer(''.join(map(str,xx)))
A174034 = lambda x: next((p for p in Primes() if is_prime(concat([nth_prime(x), nth_prime(x+1), p])))) # D. S. McNeil, Dec 02 2010

Edited and terms checked by D. S. McNeil, Dec 01 2010

A174583 a(k) is the least n such that the concatenation (n - k)"n is a prime number, for k >= 0.

Original entry on oeis.org

1, 3, 3, 7, 7, 17, 7, 9, 9, 11, 13, 17, 13, 19, 17, 19, 21, 21, 23, 27, 27, 23, 43, 33, 41, 27, 27, 29, 31, 33, 31, 33, 39, 47, 37, 39, 37, 39, 39, 41, 51, 47, 47, 61, 47, 49, 49, 53, 49, 51, 51, 59, 57, 57, 61, 57, 57, 61, 63, 63, 71, 63, 63, 67, 67, 77, 67, 69, 77, 71, 73, 77

Offset: 0

Ulrich Krug (leuchtfeuer37(AT)gmx.de), Mar 23 2010

See comments and references for A174414.
10^d*(n - k) + n has to be prime for the least d-digit n > k (k >= 0).
For (n - k)"n to be a prime, n must end in the digit 1, 3, 7, or 9.
Conjecture: a(k) = a(k+1) for an infinite number of k's.
As n > k, the number of a(k) is finite, and can be easily bounded from above.
1, 11, ... appear only once in the sequence; 3, 9, 13, 19, 21, 23, ... appear twice; 7, 17, ... three times; and so on.
Does each n that ends in the digit 1, 3, 7, or 9 appear in this sequence?
Note this interesting observation that first occurs for k = 84: 9291013 = prime(620602) = (1013 - 84)"1013, a(84) = 1013. A second example is: 9381037 = prime(626219) = (1037 - 99)"1037.
Let k be a multiple of 7, 11, or 13, then no 3-digit n exists such that (n - k)"n is prime. Proof: 10^3*(n - k) + n = n * (10^3+1) - k * 10^3 = 7 * 11 * 13 * n - k * 10^3 is not prime, as k is a multiple of 7, 11, or 13.
Similar for k-digit n with given divisors and k > 3: 10^4 + 1 = 73 * 137, 10^5 + 1 = 11 * 9091.

			11 = prime(5) = (1 - 0)"1, thus a(0) = 1.
23 = prime(9) = (3 - 1)"3, thus a(1) = 3.
13 = prime(6) = (3 - 2)"3, thus a(2) = 3.
139 = prime(34) = (39 - 38)"39, thus a(38) = 39.
9109 = prime(1130) = (109 - 100)"109, thus a(100) = 109.

Cf. A174414, A089712, A171154, A173976, A174031.

mycat := (k, n) -> parse(cat(convert(n - k, string), convert(n, string))):
sol := (k, n) -> isprime(mycat(k, n)):
a := proc(k) local n; for n from k + 1 while not sol(k, n) do od; n end:
seq(a(k), k = 0..71);  # Peter Luschny, Sep 20 2024

Edited, offset set to 0 and a(71) corrected by Peter Luschny, Sep 20 2024

A174287 Smallest natural square base q = q(k) that concatenation prime(k)//prime(k+1)//q^2 (k = 1, 2, ...) is a prime number.

Original entry on oeis.org

3, 3, 1, 11, 1, 1, 1, 1, 1, 1, 3, 7, 31, 13, 9, 1, 1, 141, 53, 37, 9, 11, 1, 7, 61, 7, 17, 13, 17, 1, 17, 11, 7, 23, 7, 27, 27, 7, 1, 9, 19, 29, 7, 29, 19, 3, 3, 1, 43, 67, 1, 7, 7, 9, 9, 1, 13, 21, 7, 7, 7, 1, 1, 43, 1, 1, 57, 1, 67, 7, 17

Offset: 1

Ulrich Krug (leuchtfeuer37(AT)gmx.de), Mar 15 2010

Note two consecutive primes prime(k)//prime(k+1)

Necessarily q is odd and has end digit 1, 3, 7 or 9

			3^2=9, 239 = prime(52) => q(1) = 3
359 = prime(72) => q(2) = 3
k=18, prime(18) = 61, 141^2 = 19881, 616719881 = prime(32151650) => q(18) = 141

J.-P. Allouche, J. Shallit: Automatic Sequences, Theory, Applications, Generalizations, Cambridge University Press, 2003

A000290, A030461, A030459, A030469, A171154, A174031, A174034

A174977 Natural numbers n = n(k) with property that string "2" appears k-times in concatenation Conc(1;n) := 1//2// ... //n (k, n = 1, 2, ...) or 0 if n(k) does not exist.

Original entry on oeis.org

2, 12, 20, 21, 0, 22, 23, 24, 25, 26, 27, 28, 29, 32, 42, 52, 62, 72, 82, 92, 102, 112, 120, 121, 0, 122, 123, 124, 125, 126, 127, 128, 129, 132, 142, 152, 162, 172, 182, 192, 200, 201, 0, 202, 203, 204, 205, 206, 207, 208, 209, 210, 211, 0, 212, 0, 220, 0, 221, 0, 0

Offset: 1

Eva-Maria Zschorn (e-m.zschorn(AT)zaschendorf.km3.de), Apr 03 2010

If a natural n(k) is made of m > 1 digits "2" then (m-1) consecutive values of n(k) = 0, preceding this n(k)

			1//2, n(1) = 2
1//...//21, n(4) = 21
n(5) does not exist, 1//...//22, n(6) = 22
1//...//29, n(13) = 29
1//...//227, n(72) = 227

K. Haase, P. Mauksch: Spass mit Mathe, Urania-Verlag Leipzig, Verlag Dausien Hanau, 2. Auflage 1985
E. I. Ignatjew, Mathematische Spielereien, Urania Verlag Leipzig/Jena/Berlin 1982

Cf. A174031, A174583.

cp's OEIS Frontend

A174414 a(n) is the smallest natural number k such that the concatenation (n+k)||k is a prime number.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Maple

PARI

Python

Formula

Extensions

A174034 The smallest prime p such that the double-concatenation prime(n) // prime(n+1) // p is a prime number.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

PARI

Sage

Extensions

A174583 a(k) is the least n such that the concatenation (n - k)"n is a prime number, for k >= 0.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Maple

Extensions

A174287 Smallest natural square base q = q(k) that concatenation prime(k)//prime(k+1)//q^2 (k = 1, 2, ...) is a prime number.

Views

Author

Keywords

Comments

Examples

References

Crossrefs

A174977 Natural numbers n = n(k) with property that string "2" appears k-times in concatenation Conc(1;n) := 1//2// ... //n (k, n = 1, 2, ...) or 0 if n(k) does not exist.

Views

Author

Keywords

Comments

Examples

References

Crossrefs