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
Examples
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
References
- 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.
Links
- Robert Israel, Table of n, a(n) for n = 1..10000
Programs
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Maple
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 -
PARI
a(n) = my(k=1); while (!isprime(eval(concat(Str(n+k), Str(k)))), k++); k; \\ Michel Marcus, Sep 17 2024
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Python
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
Formula
a(n) = 1 for n > 1 in A126785.
Extensions
Edited and corrected by Robert Israel, Sep 16 2024
Comments