cp's OEIS Frontend

A260488 Numbers of the form 2^m * (6k + 1) for m, k >= 0, and 0.

%I A260488 #38 Nov 15 2023 18:39:15
%S A260488 0,1,2,4,7,8,13,14,16,19,25,26,28,31,32,37,38,43,49,50,52,55,56,61,62,
%T A260488 64,67,73,74,76,79,85,86,91,97,98,100,103,104,109,110,112,115,121,122,
%U A260488 124,127,128,133,134,139,145,146,148,151,152,157
%N A260488 Numbers of the form 2^m * (6k + 1) for m, k >= 0, and 0.
%C A260488 Alternate definition: starting with a(0) = 0, include 2n if n is in the sequence, and include 2n+1 if no two previous terms sum to 2n+1.
%C A260488 It suffices to prove this for odd n. If n == 3(6), n-2 == 1 (mod 6); if n == 5 (mod 6), n-4 == 1 (mod 6). However, if n == 1 (mod 6), any even k in the sequence, 0 < k < n, will have k !== 0 (mod 3), and so n-k != 1 (mod 3), so it is not in the sequence; thus n must be.
%C A260488 Every nonnegative integer is the sum of two members of this sequence; every positive integer is the sum of two distinct members of this sequence. For odd n, this is by the construction in the alternate definition; and for even n, by induction n/2 = i + j, and so n = 2i + 2j.
%C A260488 It follows that:
%C A260488 * No member of this sequence except 0 is a multiple of 3.
%C A260488 * The sequence has a density of 1/3.
%C A260488 * The difference between consecutive terms is always one of {1, 2, 3, 5, 6}, and each of these occurs infinitely often, with 1 having density 1/3 and the others having density 1/6.
%C A260488 * The sequence is closed under multiplication.
%C A260488 * The primes in the sequence are A045375.
%H A260488 Franklin T. Adams-Watters, <a href="/A260488/b260488.txt">Table of n, a(n) for n = 0..10000</a>
%F A260488 n is in the sequence if and only if n = 0 or A000265(n) == 1 (mod 6). [Clarified by _Peter Munn_, Jun 11 2021]
%F A260488 n is in the sequence if n = 0 or b(n) is nonzero where b = A113448, A115235, or A123863. - _Michael Somos_, Jul 29 2015
%e A260488 Using the alternate definition:
%e A260488 1 is in the sequence because it is not the sum of 2 elements from {0}.
%e A260488 2 is in the sequence because 2 = 2*1, and 1 is in the sequence.
%e A260488 3 is not in the sequence because 3 = 1 + 2, and 1 and 2 are in the sequence.
%e A260488 6 is not in the sequence because 6 = 2*3, and 3 is not in the sequence.
%p A260488 N:= 1000: # to get all terms <= N
%p A260488 sort([0, seq(seq(2^m*(6*k+1), k = 0 .. floor((N/2^m - 1)/6)), m = 0 .. ilog2(N))]);  # _Robert Israel_, Aug 25 2015
%t A260488 mx=160;Join[{0},Sort@Flatten@Table[2^m*(6k+1),{m,0,Log2[mx]},{k,0,mx/(6*2^m)}]] (* _Robert G. Wilson v_, Aug 16 2015 *)
%o A260488 (PARI) alist(n) = my(r=vector(n),j,k);r[1]=0;j=1;while(j<n,k++;if(k\2^valuation(k,2)%6==1,r[j++]=k));r
%o A260488 (PARI) alim(n)={my(p=1,p2=p,r,j);
%o A260488   for(k=1,n,
%o A260488     if(if(k%2==0, polcoeff(p,k\2),polcoeff(p2,k)==0),p+=x^k;p2+=x^k*p));
%o A260488   r=vector(subst(p,x,1));for(k=0,n,if(polcoeff(p,k),r[j++]=k));r}
%Y A260488 Cf. A126684, A260489, primes A045375, A000265.
%Y A260488 Cf. A113448, A115235, A123863.
%K A260488 nonn,nice
%O A260488 0,3
%A A260488 _Franklin T. Adams-Watters_, Jul 27 2015