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A059555 Beatty sequence for 1 + gamma A001620.

%I A059555 #28 Feb 16 2025 08:32:44
%S A059555 1,3,4,6,7,9,11,12,14,15,17,18,20,22,23,25,26,28,29,31,33,34,36,37,39,
%T A059555 41,42,44,45,47,48,50,52,53,55,56,58,59,61,63,64,66,67,69,70,72,74,75,
%U A059555 77,78,80,82,83,85,86,88,89,91,93,94,96,97,99,100,102,104,105,107,108
%N A059555 Beatty sequence for 1 + gamma A001620.
%C A059555 Let r = gamma (the Euler constant, 0.5772...). When {k*r, k >= 1} is jointly ranked with the positive integers, A059555(n) is the position of n and A059556(n) is the position of n*r. - _Clark Kimberling_, Oct 21 2014
%H A059555 Harry J. Smith, <a href="/A059555/b059555.txt">Table of n, a(n) for n = 1..2000</a>
%H A059555 Fraenkel, Aviezri S.; Levitt, Jonathan; Shimshoni, Michael; <a href="http://dx.doi.org/10.1016/0012-365X(72)90012-X">Characterization of the set of values f(n)=[n alpha], n=1,2,...</a>, Discrete Math. 2 (1972), no. 4, 335-345.
%H A059555 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/BeattySequence.html">Beatty Sequence.</a>
%H A059555 <a href="/index/Be#Beatty">Index entries for sequences related to Beatty sequences</a>
%F A059555 a(n) = n + A038128(n).
%p A059555 A001620 := proc(n)
%p A059555         floor((1+gamma)*n) ;
%p A059555 end proc:
%p A059555 seq(A001620(n),n=1..50) ; # _R. J. Mathar_, Nov 11 2011
%t A059555 t = N[Table[k*EulerGamma, {k, 1, 200}]]; u = Union[Range[200], t]
%t A059555 Flatten[Table[Flatten[Position[u, n]], {n, 1, 100}]]  (* A059556 *)
%t A059555 Flatten[Table[Flatten[Position[u, t[[n]]]], {n, 1, 100}]] (* A059555 *)
%t A059555 (* _Clark Kimberling_, Oct 21 2014 *)
%o A059555 (PARI) { default(realprecision, 100); b=1 + Euler; for (n = 1, 2000, write("b059555.txt", n, " ", floor(n*b)); ) } \\ _Harry J. Smith_, Jun 28 2009
%o A059555 (Magma) R:=RealField(100); [Floor((1+EulerGamma(R))*n): n in [1..100]]; // _G. C. Greubel_, Aug 27 2018
%Y A059555 Beatty complement is A059556.
%K A059555 nonn,easy
%O A059555 1,2
%A A059555 _Mitch Harris_, Jan 22 2001