A282175 - OEIS

A282175

a(n) is the smallest product M=p_1*p_2*...*p_n with distinct primes p_i such that M+2^n=B, where B=q_1*q_2*...*q_n with distinct primes q_i, or a(n)=0 if there is no such M.

%I #15 Feb 28 2017 09:44:32

%S 3,6,70,2030,42978,1788710,63905142,5705962314,888081948858,

%T 120056591419170

%N a(n) is the smallest product M=p_1*p_2*...*p_n with distinct primes p_i such that M+2^n=B, where B=q_1*q_2*...*q_n with distinct primes q_i, or a(n)=0 if there is no such M.

%C Conjecturally all a(n)>0.

%C Since d(a(n)+2^n) = 2^n, where d(n) is the number of divisors of n, and d(a(n)+d(a(n)+2^n)) = d(a(n)), then it is a subsequence of sequence A175304.

%e For n=3,...,8, we have the following numbers M, B=M+2^n and their prime divisors:

%e 70 = 2 5 7; 78 = 2 3 13.

%e 2030 = 2 5 7 29; 2046 = 2 3 11 31.

%e 42978 = 2 3 13 19 29; 43010 = 2 5 11 17 23.

%e 1788710 = 2 5 7 11 23 101; 1788774 = 2 3 13 17 19 71.

%e 63905142 = 2 3 7 17 37 41 59; 63905270 = 2 5 11 13 23 29 67.

%e 5705962314 = 2 3 13 17 19 23 43 229; 5705962570 = 2 5 7 11 29 59 61 71.

%K nonn,more

%O 1,1

%A _Vladimir Shevelev_ and _Peter J. C. Moses_, Feb 07 2017

%E a(9)-a(10) from _Giovanni Resta_, Feb 28 2017