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A234541 Least k such that floor(n/k) + (n mod k) is a prime, or 0 if no such k exists.

%I A234541 #14 Sep 12 2023 12:29:49
%S A234541 0,1,1,2,1,2,1,4,2,2,1,4,1,2,3,8,1,6,1,4,2,2,1,8,2,2,5,4,1,6,1,6,2,2,
%T A234541 3,12,1,2,3,8,1,6,1,4,2,2,1,16,3,10,3,4,1,18,3,6,2,2,1,8,1,2,6,18,3,6,
%U A234541 1,4,3,4,1,14,1,2,9,4,5,6,1,16,2,2,1,12,2,2
%N A234541 Least k such that floor(n/k) + (n mod k) is a prime, or 0 if no such k exists.
%C A234541 a(n) = 1 only if n is a prime.
%C A234541 a(2m) <= m, because with k=m, floor(2m/m)+(2m mod m) = 2.
%C A234541 a(2m+1) <= 2m: floor((2m+1)/2m) + ((2m+1) mod 2m) = 1 + 1 = 2.
%H A234541 Antti Karttunen, <a href="/A234541/b234541.txt">Table of n, a(n) for n = 1..10000</a>
%o A234541 (Python)
%o A234541 primes = [2, 3]
%o A234541 primFlg = [0]*100000
%o A234541 primFlg[2] = primFlg[3] = 1
%o A234541 def appPrime(k):
%o A234541   for p in primes:
%o A234541     if k%p==0:  return
%o A234541     if p*p > k:  break
%o A234541   primes.append(k)
%o A234541   primFlg[k] = 1
%o A234541 for n in range(5, 100000, 6):
%o A234541   appPrime(n)
%o A234541   appPrime(n+2)
%o A234541 for n in range(1, 100000):
%o A234541   a = 0
%o A234541   for k in range(1, n):
%o A234541     c = n//k + n%k
%o A234541     if primFlg[c]:  # if c in primes:
%o A234541       a = k
%o A234541       break
%o A234541   print(str(a), end=', ')
%o A234541 (Scheme)
%o A234541 ;; MIT/GNU Scheme, with Aubrey Jaffer's SLIB Scheme library and function A234575bi as defined in A234575
%o A234541 (require 'factor) ;; For predicate prime? from SLIB-library.
%o A234541 (define (A234541 n) (let loop ((k 1)) (cond ((prime? (A234575bi n k)) k) ((> k n) 0) (else (loop (+ 1 k))))))
%o A234541 ;; _Antti Karttunen_, Dec 29 2013
%Y A234541 Cf. A000040, A234575.
%K A234541 nonn,easy
%O A234541 1,4
%A A234541 _Alex Ratushnyak_, Dec 27 2013