cp's OEIS Frontend

A256581 Number of conditions on m under which m^n + (m+1)^n + ... + (m+k)^n is composite for every k>=1 (see comment).

%I A256581 #30 Apr 08 2015 14:27:39
%S A256581 2,3,2,7,5,7,7,11,5,7,7,31,23,11,9,15,17,31,31,47,23,15,29,47,23,15,7,
%T A256581 15,11,31,47,95,47,15,11,127,95,47,39,63,71,63,63,95,47,31,71,95,71,
%U A256581 47,31,31,47,63,39,47,23,15,23,255,191,127,111,95,71,127
%N A256581 Number of conditions on m under which m^n + (m+1)^n + ... + (m+k)^n is composite for every k>=1 (see comment).
%C A256581 We consider a(n), n>=2, conditions of the form: all numbers P_i(m) are composite, i = 1, ..., a(n), where P_i(m) is a polynomial of power n+1. It could be proved that S_k(m)= m^n + (m+1)^n + ... + (m+k)^n, as a polynomial in m of degree n+1, is divisible by k+1. Let S*_k(m) = S_k(m)/(k+1). So we have
%C A256581 S_k(m)=S*_k(m)*(k+1)=(T_k(m)/b(n))*(k+1), (1)
%C A256581 where b(n)=A064538(n) and, by the definition of A064538, T_k(m) = b(n)*S*_k(m) is a polynomial with integer coefficients.
%C A256581 It is clear that (1) could be prime only if k+1>=2 is a divisor of b(n). In this case we should require that (1) be a composite number. We have exactly A000005(b(n))-1 such requirements. In case of n=1, a(n)=2 (see A089306, A077654).
%C A256581 Remark. Sometimes some considered conditions satisfy trivially. For example, both a(3)=2 conditions for every  m>=2 evidently hold, such that every number of the form m^3 + (m+1)^3 + ... +(m+k)^3 is composite.
%C A256581 Note that essentially this method is useful only in case of even n. Indeed, according to our comment in A001017, in case of odd n>=3 the number m^n + (m+1)^n + ... + (m+k)^n is composite for every k>=1. - _Vladimir Shevelev_, Apr 06 2015
%H A256581 Peter J. C. Moses, <a href="/A256581/b256581.txt">Table of n, a(n) for n = 1..1000</a>
%F A256581 For n>=2, a(n) = A000005(A064538(n))-1.
%Y A256581 Cf. A000005, A064538, A089306 (a(1)=2), A256385 (a(2)=3), A256546 (a(4)=7).
%K A256581 nonn
%O A256581 1,1
%A A256581 _Vladimir Shevelev_, Apr 02 2015
%E A256581 More terms from _Peter J. C. Moses_, Apr 02 2015