A230497 -id:A230497 - cp's OEIS frontend

A230498 a(n) is the minimal odd odious k>1, such that k^i, i=2,...,n, all are evil, and a(n)=0, if there is no such k.

13, 21, 21, 47, 265, 607, 883, 883, 883, 883, 10865, 10865, 58241, 58241, 58241, 75781, 367815, 766165, 2931371, 5288671, 5288671, 14838843, 14838843, 14838843, 33417397, 737812313, 2774333869, 3513898753, 3513898753, 3513898753, 14369883465, 14369883465, 22865025261

Offset: 2

Vladimir Shevelev and Peter J. C. Moses, Oct 21 2013

A conjugate sequence to A230495 and A230496.

Conjecture: For all n, a(n) > 0.

Amiram Eldar, Table of n, a(n) for n = 2..39

Cf. A000069, A001969, A230494, A230495, A230496, A230497.

evQ[n_] := EvenQ[DigitCount[n, 2, 1]]; evExp[n_] := Module[{e = 1, p = n^2}, If[! evQ[n], While[evQ[p], p *= n; e++]]; e]; seq[nmax_] := Module[{e, emax = 1, n = 3, s = {}}, Do[e = evExp[n]; If[e > emax, s = Join[s, ConstantArray[n, e - emax]]; emax = e], {n, 3, nmax, 2}]; s]; seq[11000] (* Amiram Eldar, Aug 03 2023 *)

a(27)-a(34) from Amiram Eldar, Aug 03 2023

A230500 Indices of "power odious-evil stability" of odd integers 2*n-1 > 1 (see comment).

Original entry on oeis.org

3, 1, 2, 1, 2, 1, 3, 1, 2, 1, 1, 3, 4, 1, 2, 1, 3, 1, 1, 2, 1, 2, 1, 3, 1, 3, 2, 4, 4, 1, 3, 1, 3, 1, 1, 1, 1, 1, 3, 1, 3, 1, 2, 1, 2, 1, 1, 3, 2, 1, 8, 4, 1, 1, 2, 1, 1, 3, 2, 2, 2, 1, 2, 1, 3, 1, 1, 1, 1, 3, 2, 2, 1, 1, 1, 1, 1, 3, 3, 1, 1, 2, 3, 1, 1, 3, 1

Offset: 2

Vladimir Shevelev, Oct 21 2013

We say that odious 2*n-1, n>=2, has "power odious-evil stability" of index k, if k is the maximal, such that the numbers 2*n-1, (2n-1)^2, ... , (2n-1)^k all are evil or all are odious.

For example, 3, 3^2, 3^3 are evil, while 3^4 is odious. So 3 has power stability of index 3. For 5 it is 1, for 7 it is 2, etc.

Peter J. C. Moses, Table of n, a(n) for n = 2..10001
Vladimir Shevelev, "Odious-evil stability" of integers

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

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

cp's OEIS Frontend

A230498 a(n) is the minimal odd odious k>1, such that k^i, i=2,...,n, all are evil, and a(n)=0, if there is no such k.

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A230500 Indices of "power odious-evil stability" of odd integers 2*n-1 > 1 (see comment).

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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).

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cp's OEIS Frontend

A230498 a(n) is the minimal odd odious k>1, such that k^i, i=2,...,n, all are evil, and a(n)=0, if there is no such k.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Extensions

A230500 Indices of "power odious-evil stability" of odd integers 2*n-1 > 1 (see comment).

Views

Author

Keywords

Comments

Links

Crossrefs

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

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).