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A054906 Smallest number x such that sigma(x+2n) = sigma(x)+2n (first definition).

%I A054906 #36 Feb 06 2017 03:19:46
%S A054906 3,3,5,3,3,5,3,3,5,3,7,5,3,3,7,5,3,5,3,3,5,3,7,5,3,7,5,3,3,7,5,3,5,3,
%T A054906 3,7,5,3,5,3,7,5,3,13,7,5,3,5,3,3,5,3,3,5,3,19,13,11,13,7,5,3,5,3,7,5,
%U A054906 3,3,11,11,7,5,3,3,7,5,3,7,5,3,5,3,7,5,3,7,5,3,3,11,11,7,5,3,3,5,3,3,13
%N A054906 Smallest number x such that sigma(x+2n) = sigma(x)+2n (first definition).
%C A054906 Least (prime) solutions for phi(x+2n)=phi(x)+2n seems to be identical to this sequence, while prime solutions are indeed identical to this sequence.
%C A054906 2nd definition = smallest number x such that phi(x+2n)=phi(x)+2n.
%C A054906 3rd definition = smallest primes p such that p+2n=q prime (A020483).
%C A054906 The 3 definitions are identical or conjectured to be identical.
%C A054906 The definitions are not identical if we do not take the smallest numbers. These smallest solutions are believed to be always prime numbers.
%C A054906 Duplicate of A020483, assuming that the 3rd definition is also correct. - _R. J. Mathar_, Apr 26 2015
%C A054906 If it can be proved that all these definitions are identical, then this entry should be merged with A020483. - _N. J. A. Sloane_, Feb 06 2017
%D A054906 Sivaramakrishnan,R.(1989):Classical Theory of Arithmetical Functions. Marcel Dekker,Inc., New York.
%H A054906 Michael De Vlieger, <a href="/A054906/b054906.txt">Table of n, a(n) for n = 1..10000</a>
%F A054906 Minimal solutions to A000203(x+2n)=A000203(x)+2n or to A000010(x+2n)=A000010(x)+2n or to p+2n=q; p, q primes, a(n)=p.
%F A054906 a(n) <= A054905(n). - _R. J. Mathar_, Apr 28 2015
%e A054906 n-th primes 2,3,5,7,11,13, are solutions to sigma(x+2n)=2n+sigma(x) at 2n=2,6,22,116,88.
%p A054906 A054906 := proc(n)
%p A054906     local x;
%p A054906     for x from 0 do
%p A054906         if numtheory[sigma](x+2*n) = numtheory[sigma](x)+2*n then
%p A054906             return x;
%p A054906         end if;
%p A054906     end do:
%p A054906 end proc:
%p A054906 seq(A054906(n),n=1..40); # _R. J. Mathar_, Sep 23 2016
%t A054906 Table[x = 1; While[DivisorSigma[1, x + 2 n] != DivisorSigma[1, x] + 2 n, x++]; x, {n, 100}] (* _Michael De Vlieger_, Feb 05 2017 *)
%o A054906 (PARI) a(n) = my(x = 1); while(sigma(x+2*n) != sigma(x)+2*n, x++); x; \\ _Michel Marcus_, Dec 17 2013
%Y A054906 Cf. A023200-A023203, A015913-A015917, A000203, A000010, A020483.
%K A054906 nonn
%O A054906 1,1
%A A054906 _Labos Elemer_, May 23 2000