cp's OEIS Frontend

A275237 Smallest number k > 0 such that sigma(x) and sigma(x)+2 are both prime, where x = (6k+1)^(6n+4), or -1 if no such k exists.

%I A275237 #37 Aug 19 2016 04:32:44
%S A275237 1,348,436,6018,5880,-1,4612,26921,16166,81111,-1,426260,-1,181876,
%T A275237 227180,-1,12836,287388,2317,-1,-1,1128403,668927,-1,5295,-1,-1,
%U A275237 490118,2217967,1607226,-1,1212183,100728,-1,-1,-1,-1,1191713,43475567,165965,-1,2915491,361885,4159496,3398061,-1,88930,-1,10451327,-1,-1
%N A275237 Smallest number k > 0 such that sigma(x) and sigma(x)+2 are both prime, where x = (6k+1)^(6n+4), or -1 if no such k exists.
%C A275237 If x is a number such that sigma(x) and sigma(x)+2 are both prime (A274962), then x = 2 or x is of the form (6k+1)^(6r+4) where 6k+1 is prime.
%C A275237 For p = 6*k+1, sigma(p^34) = (46656*k^6 + 54432*k^5 + 27216*k^4 + 7560*k^3 + 1260*k^2 + 126*k + 7) * (1296*k^4 + 1080*k^3 + 360*k^2 + 60*k + 5) * c(k), thus a(5) = -1. - _Altug Alkan_ , Jul 21 2016
%C A275237 Similarly a(12) = a(19) = a(23) = a(26) = a(33) = a(34) = -1. Furthermore, for all r > 0, a(5*r) = -1 since sigma((6k+1)^(30r+4)) = ((6*k+1)^(6*r) + ((6*k+1)^(6*r) -1)/(6*k))*(1296*k^4*(6*k + 1)^(24*r) + 864*k^3*(6*k + 1)^(24*r) + 216*k^3*(6*k + 1)^(18*r) + 216*k^2*(6*k + 1)^(24*r) + 108*k^2*(6*k + 1)^(18*r) + 36*k^2*(6*k + 1)^(12*r) + 24*k*(6*k + 1)^(24*r) + 18*k*(6*k + 1)^(18*r) + 12*k*(6*k + 1)^(12*r) + 6*k*(6*k + 1)^(6*r) + (6*k + 1)^(24*r) + (6*k + 1)^(18*r) + (6*k + 1)^(12*r) + (6*k + 1)^(6*r) + 1). - _Chai Wah Wu_, Jul 21 2016
%F A275237 a(A059324(n)) = -1. - _Altug Alkan_, Aug 13 2016
%e A275237 For n = 0, x = 7^4 is the smallest fourth power such that sigma(x) and sigma(x)+2 are both prime, thus a(0) = 1.
%Y A275237 Cf. A000203, A023194, A249763, A274962, A274963.
%K A275237 sign,hard
%O A275237 0,2
%A A275237 _Chai Wah Wu_, Jul 20 2016
%E A275237 a(31)-a(37) from _Chai Wah Wu_, Aug 01 2016
%E A275237 a(38)-a(50) from _Chai Wah Wu_, Aug 18 2016