cp's OEIS Frontend

A178102 2^(absolute difference between prime factors of n-th semiprime) mod (n-th semiprime).

%I A178102 #22 Jun 27 2024 22:15:39
%S A178102 1,2,1,8,4,4,16,6,1,20,25,26,4,10,10,12,1,13,9,43,44,16,61,52,56,16,
%T A178102 62,16,22,22,64,70,24,44,80,28,59,30,72,1,92,31,97,106,34,106,36,4,
%U A178102 136,110,64,40,40,9,42,1,133,134,46,81,64,146,151,152,121
%N A178102 2^(absolute difference between prime factors of n-th semiprime) mod (n-th semiprime).
%C A178102 From _Robert Israel_, Apr 05 2020: (Start)
%C A178102 If A001358(n) = 2*p, then a(n) = (p+1)/2 if p == 3 (mod 4), or (3*p+1)/2 if p == 1 (mod 4).
%C A178102 If A001358(n) = 3*p with p > 3, then a(n) = (3*p+1)/4 if p == 1 (mod 4), or (9*p+1)/4 if p == 3 (mod 4). (End)
%H A178102 Robert Israel, <a href="/A178102/b178102.txt">Table of n, a(n) for n = 1..10000</a>
%e A178102 a(1)=1 because the first semiprime is 4=2*2 and 2^(2-2) mod 4 = 1.
%e A178102 a(11)=25 because the 11th semiprime is 33=3*11 and 2^(11-3) mod 33 = 25.
%p A178102 b:= proc(n) option remember;
%p A178102   local k;
%p A178102   if n=1 then 4
%p A178102          else for k from b(n-1)+1 while
%p A178102                   isprime(k) or add (i[2], i=ifactors(k)[2])<>2
%p A178102               do od; k
%p A178102   fi
%p A178102 end:
%p A178102 a:= proc(n)
%p A178102   local l;
%p A178102   l:= ifactors (b(n))[2];
%p A178102   if nops (l)=1 then 1
%p A178102   else 2 &^ abs(l[1][1]-l[2][1]) mod b(n)
%p A178102   fi
%p A178102 end:
%p A178102 seq (a(n), n=1..65);
%t A178102 Mod[2^Differences[FactorInteger[#][[All,1]]],#]&/@Select[Range[300], PrimeOmega[ #] == 2&]/.{}->1//Flatten (* _Harvey P. Dale_, Dec 25 2018 *)
%Y A178102 Cf. A000079, A001358.
%K A178102 nonn,look
%O A178102 1,2
%A A178102 _Juri-Stepan Gerasimov_, Dec 16 2010
%E A178102 Edited by _Alois P. Heinz_, Dec 17 2010