A224072 -id:A224072 - cp's OEIS frontend

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

Vladimir Shevelev, Sep 30 2013

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.

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.

Cf. A001969, A000069, A224072.

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

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

A230499 a(n) is the maximal number k of consecutive numbers of the form (2n-1)(2*i-1), i=1,2,...,k, which are all evil or all odious (A000069, A001969).

Original entry on oeis.org

1, 3, 2, 4, 4, 1, 1, 9, 8, 1, 1, 1, 1, 1, 1, 16, 16, 1, 1, 1, 1, 3, 4, 1, 1, 3, 2, 1, 1, 1, 1, 33, 32, 1, 1, 1, 1, 2, 2, 1, 1, 2, 2, 2, 2, 2, 5, 1, 1, 5, 2, 4, 20, 4, 3, 1, 1, 4, 2, 1, 1, 1, 1, 64, 64, 1, 1, 1, 1, 2, 4, 1, 1, 3, 4, 5, 4, 3, 2, 1, 1, 2, 4, 3, 3, 1, 1

Offset: 1

Vladimir Shevelev, Oct 21 2013

We call a(n) the multiplicative index of odious-evil stability of 2*n-1.

			For n=2, t=2*n-1=3. We see that 3*1=3, 3*3=9,3*5=15 are evil, but 3*7=21 is odious. So, a(2)=3.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
V. Shevelev, "Odious-evil stability" of integers

Cf. A000069, A001969, A230454, A230495, A230496, A230497, A230498, A000225, A224072.

a(n)=my(t=2*n-1,H=hammingweight(t)%2,i=3); while(H == hammingweight(i*t)%2, i+=2); i\2 \\ Charles R Greathouse IV, Oct 22 2013

If 2*n-1 is Mersenne number (A000225), then a(n)>=n; if 2*n-1 is odious such that 6*n-3 is not in A224072, then a(n)=1.

a(17)-a(87) from Charles R Greathouse IV, Oct 22 2013

A230456 Odd evil a(n) (A001969), such that 3*a(n) and a(n)+3 are odious (A000069).

Original entry on oeis.org

23, 29, 39, 71, 95, 119, 125, 135, 159, 263, 287, 343, 349, 359, 373, 383, 407, 413, 423, 437, 469, 479, 503, 509, 519, 543, 599, 605, 615, 629, 639, 663, 669, 679, 711, 741, 791, 797, 807, 839, 869, 917, 933, 1031, 1055, 1111, 1117, 1127, 1141, 1151, 1175

Offset: 1

Vladimir Shevelev, Oct 19 2013

Or a(n) is odd number such that the polynomial x^2 - (a(n)+3)*x + 3*a(n) has odious coefficients and evil roots.

3*a(n) is in the A224072.

Cf. A000069, A001969, A224072.

evilQ[n_]:=EvenQ[DigitCount[n,2][[1]]]; odiousQ[n_]:=OddQ[DigitCount[n,2][[1]]]; Select[Range[3000], OddQ[#] && evilQ[#] && odiousQ[3#] && odiousQ[#+3]&] (* Peter J. C. Moses, Oct 19 2013 *)

More terms from Peter J. C. Moses, Oct 19 2013

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

A230499 a(n) is the maximal number k of consecutive numbers of the form (2n-1)(2*i-1), i=1,2,...,k, which are all evil or all odious (A000069, A001969).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Formula

Extensions

A230456 Odd evil a(n) (A001969), such that 3*a(n) and a(n)+3 are odious (A000069).

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

Extensions

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

A230499 a(n) is the maximal number k of consecutive numbers of the form (2*n-1)*(2*i-1), i=1,2,...,k, which are all evil or all odious (A000069, A001969).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Formula

Extensions

A230456 Odd evil a(n) (A001969), such that 3*a(n) and a(n)+3 are odious (A000069).

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

Extensions

A230499 a(n) is the maximal number k of consecutive numbers of the form (2n-1)(2*i-1), i=1,2,...,k, which are all evil or all odious (A000069, A001969).