A079498 - OEIS

%I #23 May 29 2022 18:00:14

%S 0,5,7,11,13,15,19,21,26,29,31,33,37,39,44,45,50,52,55,57,62,63,68,70,

%T 74,76,78,83,85,87,91,93,98,99,104,106,109,111,116,117,122,124,128,

%U 130,132,135,140,142,146,148,150,154,156,161,163,165,170,171,176,178,182,184

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

%C 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.

%C 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

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

%H Amiram Eldar, <a href="/A079498/b079498.txt">Table of n, a(n) for n = 1..10000</a>

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

%t 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]

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

%K base,easy,nonn

%O 1,2

%A _Carlos Alves_, Jan 21 2003

%E a(1) = 0 inserted and offset corrected by _Amiram Eldar_, Jan 05 2020