A079498 Numbers whose sum of digits in base b gives 0 (mod b), for b = 3.

0, 5, 7, 11, 13, 15, 19, 21, 26, 29, 31, 33, 37, 39, 44, 45, 50, 52, 55, 57, 62, 63, 68, 70, 74, 76, 78, 83, 85, 87, 91, 93, 98, 99, 104, 106, 109, 111, 116, 117, 122, 124, 128, 130, 132, 135, 140, 142, 146, 148, 150, 154, 156, 161, 163, 165, 170, 171, 176, 178, 182, 184

Offset: 1

Author

Carlos Alves, Jan 21 2003

Keywords

• base
• easy
• nonn

Comments

In base 2 this gives the "Evil Numbers" (cf. A001969). One may conjecture that in base b the asymptotic slope will be b and asymptotic density 1/b for each result (mod b). Cases b=31 or b=61 gave considerable number of primes on the sequence.

Proof of this conjecture: in general, the sequence d with terms d(n) = sum of digits of n written in base b (mod b) is a fixed point of the generalized Thue-Morse morphism 0->01..b-1, 1->12..0, etc. See A053839 for the case b=4. It follows directly from this that all symbols have asymptotic density 1/b, and therefore that the positional sequences all have asymptotic slope b. - Michel Dekking, Apr 18 2019

Positions of 0's in A053838. Cf. A026601.

Examples

			83 is a term since 83 = (1,0,0,0,2)_3 and 1 + 0 + 0 + 0 + 2 = 3 == 0 (mod 3).
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A001969. See A053840 for base b=5. See A141803 for an array with all b.

Programs

Mathematica

Ev = Function[{b, x}, vx = IntegerDigits[x, b]; Mod[Apply[Plus, vx], b]]; Seq = Function[{b, n}, Flatten[Position[Table[Ev[b, k], {k, 0, n}], 0]] - 1]; sb = Seq[3, 1000]

Extensions

a(1) = 0 inserted and offset corrected by Amiram Eldar, Jan 05 2020