Search: id:A071537
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%I A071537
%S A071537 1,4,6,8,10,9,14,12,15,16,22,18,26,21,20,24,34,25,38,27,28,33,46,30,32,
               39,35,36,58,
%T A071537 40,62,42,44,51,45,48,74,57,52,49,82,50,86,55,54,69,94,56,60,63,68,65,
               106,64,66,70,
%U A071537 76,87,118,72,122,93,75,77,78,80,134,85,92,84,142,81,146,111,88,95,90,
               91,158,96,98
%N A071537 Least integer m >= n such that there exist integers k_n, k_n+1, ..., 
               k_m with nonzero k_n, for which n^(k_n) * (n+1)^(k_n+1) * ... * m^(k_m) 
               = 1.
%C A071537 Conjecture: this is a permutation of the nonprimes A018252.
%C A071537 Comment from Max Alekseyev (start): The conjecture is true. It is easy 
               to see that (i) a(n)=m cannot be prime; (ii) k_m is always nonzero; 
               (iii) for any nonprime s, there exist integers k_p,...,k_s, where 
               p is the smallest prime divisor of s, k_p and k_s are nonzero, for 
               which p^(k_p)*...*s^(k_s) = 1. The bijectivity of a(n) (as a function 
               between positive integers and nonprimes) now follows from the lemma.
%C A071537 *Lemma*. If a < b <= m and a^(p_a)*...* m^(p_m) = b^(q_b)*...* m^(q_m) 
               = 1 where p_a, p_m, q_b and q_m are nonzero, then there exists integer 
               s, a <= s < m, such that a^(r_a)*...* m^(r_s) = 1 where r_a and r_s 
               are nonzero. Similarly, if a <= s < m and a^(p_a)*...* s^(p_s) = 
               a^(q_a)*...* m^(q_m) = 1 where p_a, p_s, q_a and q_m are nonzero, 
               then there exists integer b, a < b <= m, such that b^(r_b)*...* m^(r_m) 
               = 1 where r_b and r_m are nonzero. (end)
%e A071537 a(18) = 25: we have 18^4 * 19^0 * 20^10 * 21^0 * 22^0 * 23^0 * 24^-8 
               * 25^-5 = 1.
%o A071537 (PARI) { a(n) = local(m,d,f,M,M2); if(n==1,return(1)); m=n; while(1, 
               m++; if(isprime(m),next); d=primepi(m); M=matrix(d,m-n+1); for(k=0,
               m-n, f=factor(n+k); for(j=1,matsize(f)[1], M[primepi(f[j,1]),k+1]=f[j,
               2] )); M2=matrix(d,m-n,i,j,M[i,j+1]); if(matrank(M)==matrank(M2),
               return(m)); ) } [From Max Alekseyev (maxale(AT)gmail.com), Oct 22 
               2008]
%Y A071537 Sequence in context: A144123 A132088 A162600 this_sequence A161396 A145256 
               A087789
%Y A071537 Adjacent sequences: A071534 A071535 A071536 this_sequence A071538 A071539 
               A071540
%K A071537 nice,nonn
%O A071537 1,2
%A A071537 Alex Fink (fink(AT)cadvision.com), May 30 2002
%E A071537 Corrected definition, comments and extra terms from Max Alekseyev (maxale(AT)gmail.com), 
               Oct 22 2008

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