login
Smallest number x such that sigma(x+2n) = sigma(x)+2n (first definition).
7

%I #36 Feb 06 2017 03:19:46

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

%C 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 2nd definition = smallest number x such that phi(x+2n)=phi(x)+2n.

%C 3rd definition = smallest primes p such that p+2n=q prime (A020483).

%C The 3 definitions are identical or conjectured to be identical.

%C The definitions are not identical if we do not take the smallest numbers. These smallest solutions are believed to be always prime numbers.

%C Duplicate of A020483, assuming that the 3rd definition is also correct. - _R. J. Mathar_, Apr 26 2015

%C 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 Sivaramakrishnan,R.(1989):Classical Theory of Arithmetical Functions. Marcel Dekker,Inc., New York.

%H Michael De Vlieger, <a href="/A054906/b054906.txt">Table of n, a(n) for n = 1..10000</a>

%F 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 a(n) <= A054905(n). - _R. J. Mathar_, Apr 28 2015

%e 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 := proc(n)

%p local x;

%p for x from 0 do

%p if numtheory[sigma](x+2*n) = numtheory[sigma](x)+2*n then

%p return x;

%p end if;

%p end do:

%p end proc:

%p seq(A054906(n),n=1..40); # _R. J. Mathar_, Sep 23 2016

%t 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 (PARI) a(n) = my(x = 1); while(sigma(x+2*n) != sigma(x)+2*n, x++); x; \\ _Michel Marcus_, Dec 17 2013

%Y Cf. A023200-A023203, A015913-A015917, A000203, A000010, A020483.

%K nonn

%O 1,1

%A _Labos Elemer_, May 23 2000