A112545 - OEIS

%I #12 Apr 22 2017 13:12:50

%S 7,5,2,5,7,2,5,3,3,3,3,5,11,3,53,3,3,3,5,3,3,3,3,5,5,13,53,5,59,61,3,

%T 3,11,5,3,157,3,3,173,3,5,11,97,7,3,211,3,113,5,3,3,5,3,257,263,3,3,3,

%U 5,7,5,151,5,157,7,3,3,7,5,3,3,3,373,3,3,3,5,13,5,5,5,7,3,3,3,3,5,5,29,3,3

%N Least odd number k greater than 1 such that the sum of the predecessor and successor primes of the n-th prime is divisible by k or if no such odd k exists then 2.

%C From _Robert Israel_, Apr 20 2017: (Start)

%C a(n) = A078701(prime(n-1)+prime(n+1)) unless that is 1, in which case a(n)=2.

%C a(n) = 2 if and only if for some m, A007053(m) = n or n-1 with prime(n-1)+prime(n+1) = 2^(m+1). The first m for which this occurs are 3,4,9,379,593, corresponding to n = 4,7,97 and approximately 3*10^116 and 1*10^181.  Are there infinitely many? (End)

%H Robert Israel, <a href="/A112545/b112545.txt">Table of n, a(n) for n = 2..10000</a>

%p f:= proc(n) local t; t:= min(numtheory:-factorset(ithprime(n-1)+ithprime(n+1)) minus {2}); if t::integer then t else 2 fi end proc:

%p map(f, [$2..200]); # _Robert Israel_, Apr 20 2017

%t f[n_] := Block[{k = 3, s = Prime[n - 1] + Prime[n + 1]}, While[Mod[s, k] != 0 && k <= s, k += 2]; If[k > s, 2, k]]; Table[ f[n], {n, 2, 92}]

%o (PARI) a(n) = {p = prime(n); s = precprime(p-1) + nextprime(p+1); f = factor(s); if (#f~ > 1, f[2,1], f[1,1]);} \\ _Michel Marcus_, Apr 22 2017

%Y Cf. A000040, A112686, A112681, A112794, A112731, A112789, A112795, A112796, A112804, A112847, A112859, A113155, A113156, A113157, A113158.

%Y Cf. A007053, A078701.

%K nonn

%O 2,1

%A _Robert G. Wilson v_, Jan 11 2006