A147568 a(n) = 2*A000695(n)+3.

3, 5, 11, 13, 35, 37, 43, 45, 131, 133, 139, 141, 163, 165, 171, 173, 515, 517, 523, 525, 547, 549, 555, 557, 643, 645, 651, 653, 675, 677, 683, 685, 2051, 2053, 2059, 2061, 2083, 2085, 2091, 2093, 2179, 2181, 2187, 2189, 2211, 2213, 2219, 2221, 2563, 2565, 2571

Offset: 0

Author

Vladimir Shevelev, Nov 07 2008

Keywords

• nonn

Comments

Every odd number m>=9 is a unique sum of the form a(k)+2a(l); moreover this sequence is the unique one with such property. In connection with A103151, note that there is no subsequence T of primes such that every odd number m>=9 is expressible as a unique sum of the form m=p+2q, where p and q are in T. One can prove that if one replaces 9 by any integer x_o>9, the statement remains true (see the Shevelev link).

Links

• Vladimir Shevelev, On Unique Additive Representations of Positive Integers and Some Close Problems, arXiv:0811.0290 [math.NT], 2008.

Crossrefs

Cf. A000695, A062880, A103151.

Programs

Mathematica

(* b = A000695 *) b[n_] := If[n==0, 0, If[EvenQ[n], 4 b[n/2] , b[n-1]+1]];
a[n_] := 2 b[n] + 3; Table[a[n], {n, 0, 50}] (* Jean-François Alcover, Dec 14 2018 *)

PARI

a000695(n) = fromdigits(binary(n), 4);
a(n) = 2*a000695(n)+3; \\ Michel Marcus, Dec 13 2018

Extensions

More terms from Michel Marcus, Dec 13 2018