A229826 Evil (A001969) numbers divisible by 7 but not divisible by 3.

77, 119, 154, 175, 238, 245, 287, 308, 329, 343, 350, 371, 413, 427, 455, 469, 476, 490, 497, 553, 574, 581, 616, 658, 679, 686, 700, 742, 763, 791, 826, 833, 854, 910, 917, 931, 938, 952, 980, 994, 1043, 1085, 1106, 1127, 1141, 1148, 1162, 1169, 1232, 1253

Offset: 1

Author

Vladimir Shevelev, Sep 30 2013

Keywords

• nonn
• base

Comments

By the Moser-Newman phenomenon, among the first N positive integers divisible by 3, the evil numbers are always in the majority. But what happens if we remove from the positive numbers the multiples of 3? We conjecture that in this case we obtain another phenomenon: among the first N such positive integers divisible by 7, the odious numbers (A000069) are always in the majority.

Links

• Harvey P. Dale, Table of n, a(n) for n = 1..1000
• J. Coquet, A summation formula related to the binary digits, Inventiones Mathematicae 73 (1983), pp. 107-115.
• D. J. Newman, On the number of binary digits in a multiple of three, Proc. Amer. Math. Soc. 21 (1969) 719-721.
• Vladimir Shevelev, Generalized Newman phenomena and digit conjectures on primes, Internat. J. of Mathematics and Math. Sciences, 2008 (2008), Article ID 908045, 1-12.

Crossrefs

Cf. A001969, A000069, A224072.

Programs

Mathematica

With[{evil=Select[Range[0,1500],EvenQ[DigitCount[#,2,1]]&]},Select[evil, Divisible[#,7]&&!Divisible[#,3]&]] (* Harvey P. Dale, Dec 04 2014 *)

PARI

is(n)=hammingweight(n)%2==0 && gcd(n,21)==7 \\ Charles R Greathouse IV, Sep 30 2013