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A181642 Minimal sequence whose forwards van Eck transform is the sequence of prime numbers. 2


%S 0,1,0,2,1,3,4,0,5,6,2,7,8,9,10,1,11,12,3,13,14,15,16,4,17,18,0,19,20,

%T 21,22,5,23,24,25,26,27,28,6,29,30,2,31,32,33,34,35,36,7,37,38,39,40,

%U 8,41,42,9,43,44,45,46,10,47,48,49,50,51,52,1,53,54

%N Minimal sequence whose forwards van Eck transform is the sequence of prime numbers.

%C At each step, the minimum available integer is used.

%C From _Rémy Sigrist_, Aug 12 2017: (Start)

%C a(n)=0 iff n belongs to A074271.

%C a(n)=1 iff n > 1 and n belongs to A259408.

%C For any k > 0, A064427(k) = least n such that a(n) = k-1.

%C (End)

%H Rémy Sigrist, <a href="/A181642/b181642.txt">Table of n, a(n) for n = 1..10000</a>

%e a(1)=0. Next 0 is at distance 2 (1st prime): a(3)=0.

%e a(2)=1. Next 1 is at distance 3 (2nd prime): a(5)=1.

%e a(3)=0. Next 0 is at distance 5 (3rd prime): a(8)=0.

%e For a(4), we can use neither 0 (distance 1 from previous 0 would lead to an incongruence) nor 1 (distance 1 from subsequent 1 would lead to another incongruence). Therefore we must use 2.

%e Next 2 must be at distance 7 (4th prime): a(11)=2. And so on.

%p P:=proc(q,h) local i,k,n,t,x; x:=array(1..h); for k from 1 to h do x[k]:=-1; od; x[1]:=0; i:=0; t:=0;for n from 1 to q do if isprime(n) then i:=i+1; if x[i]>-1 then x[i+n]:=x[i]; else t:=t+1; x[i]:=t; x[i+n]:=x[i]; fi; fi; od; seq(x[k],k=1..79); end: P(400,500);

%o (PARI) a = vector(71, i, -1); u = 0; for (n=1, #a, if (a[n]<0, o = n; while (o <= #a, a[o] = u; o += prime(o)); u++); print1 (a[n] ", ")) \\ _Rémy Sigrist_, Aug 12 2017

%Y Cf. A000040, A064427, A074271, A181391, A259408.

%K easy,nonn,changed

%O 1,4

%A _Paolo P. Lava_ & _Giorgio Balzarotti_, Nov 03 2010

%E More terms from _Rémy Sigrist_, Aug 12 2017

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Last modified June 16 21:20 EDT 2019. Contains 324155 sequences. (Running on oeis4.)