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A248835 a(n) = n + A033677(n), where A033677(n) is the smallest divisor of n >= sqrt(n).

%I A248835 #38 Sep 23 2023 03:39:51
%S A248835 2,4,6,6,10,9,14,12,12,15,22,16,26,21,20,20,34,24,38,25,28,33,46,30,
%T A248835 30,39,36,35,58,36,62,40,44,51,42,42,74,57,52,48,82,49,86,55,54,69,94,
%U A248835 56,56,60,68,65,106,63,66,64,76,87,118,70,122,93
%N A248835 a(n) = n + A033677(n), where A033677(n) is the smallest divisor of n >= sqrt(n).
%C A248835 Conjecture: For n > 1 there is at least one prime in [n, a(n)] exclusive.
%C A248835 a(n) = 2*n when n is prime.
%C A248835 When n = A002620(m), then a(n) = A002620(m+1), i.e., quarter-squares. Oppermann's conjecture states that there is at least one prime in [A002620(m), A002620(m+1)] exclusive.
%C A248835 When n is square, repeated values for a(n) occur at n-1 and n. These values are A002378(sqrt(n)), i.e., oblong numbers.
%C A248835 When n = A002378(m), then a(n) = (m+1)^2.
%H A248835 Wikipedia, <a href="http://en.wikipedia.org/wiki/Oppermann%27s_conjecture">Oppermann's conjecture</a>
%e A248835 When n = 40, the smallest divisor of 40 that is greater than or equal to sqrt(40) is 8 so a(40)=48.
%t A248835 a248835[n_Integer] := n + Min[Select[Divisors[n], # >= Sqrt[n] &]]; a248835 /@ Range[120] (* _Michael De Vlieger_, Nov 10 2014 *)
%o A248835 (Sage)
%o A248835 [n+min([x for x in divisors(n) if x>=sqrt(n)]) for n in [1..100]] # _Tom Edgar_, Oct 15 2014
%o A248835 (PARI) a(n)=fordiv(n,d,if(d^2>=n,return(n+d))) \\ _Charles R Greathouse IV_, Oct 21 2014
%Y A248835 Cf. A002378, A002620, A033677.
%K A248835 nonn
%O A248835 1,1
%A A248835 _Bob Selcoe_, Oct 15 2014