A230454 -id:A230454 - cp's OEIS frontend

A230495 a(n) is the minimal odd evil k, such that k^i, i=1,2,...,n, all are evil, and a(n)=0, if there is no such k.

3, 3, 3, 27, 287, 287, 287, 287, 783, 783, 783, 19099, 20249, 34391, 80577, 92589, 211183, 211183, 211183, 1995137, 4270443, 4270443, 4270443, 4270443, 17026791, 317108969, 317108969, 317108969, 979104339, 979104339, 6044000725, 21911775681, 21911775681, 26576734759

Offset: 1

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

We conjecture that a(n)>0 for all n.

Vladimir Shevelev, "Odious-evil stability" of integers, post to the SeqFan Mailing List, Oct 20 2013.

Cf. A000069, A001969, A230454.

fQ[n_] := EvenQ[DigitCount[n, 2,1]]; a[n_] := Module[{k=3}, While[ LengthWhile[ Range[n], fQ[k^#] &] != n, k+=2]; k]; Array[a, 12] (* Amiram Eldar, Dec 10 2018 *)

isevil(n) = (hammingweight(n) % 2) == 0; \\ A000069
isok(k, n) = {if (!isevil(k), return (0)); for (i=1, n, if (!isevil(k^i), return (0));); return (1);}
a(n) = {my(k=1); while(!isok(k,n), k += 2); k;}

a(26) from Michel Marcus, Dec 10 2018

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

A230496 a(n) is the minimal odd odious k > 1, such that k^i, i=1,2,...,n, all are odious, or a(n)=0, if there is no such k.

Original entry on oeis.org

7, 7, 25, 59, 103, 103, 103, 103, 1305, 1305, 10525, 10525, 10525, 25127, 25127, 25127, 106501, 215735, 810163, 810163, 810163, 9554677, 13101403, 13101403, 14299679, 37795511, 37795511, 37795511, 3197105709, 3197105709, 5386711727, 14904706741, 20696039773

Offset: 1

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

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

Vladimir Shevelev, "Odious-evil stability" of integers, post to the SeqFan Mailing List, Oct 20 2013.

Cf. A000069, A001969, A230454, A230495.

odQ[n_] := OddQ[DigitCount[n, 2, 1]]; odExp[n_] := Module[{e = 1, p = n}, If[odQ[n], While[odQ[p], p *= n; e++]]; e]; seq[nmax_] := Module[{e, emax = 1, n = 3, s = {}}, Do[e = odExp[n]; If[e > emax, s = Join[s, ConstantArray[n, e - emax]]; emax = e], {n, 3, nmax, 2}]; s]; seq[26000] (* Amiram Eldar, Aug 03 2023 *)

a(17)-a(22) from Alois P. Heinz, Oct 21 2013
a(23)-a(27) from Peter J. C. Moses, Oct 22 2013
a(28)-a(33) from Amiram Eldar, Aug 03 2023

A230497 a(n), n>=2, is the minimal odd evil k, such that k^i, i=2,3,...,n, all are odious, and a(n)=0, if there is no such k.

Original entry on oeis.org

5, 23, 71, 85, 89, 163, 225, 225, 225, 225, 6075, 6075, 9859, 9859, 9859, 9859, 9859, 9859, 5031037, 10430265, 11896187, 11896187, 11896187, 22402429, 340713205, 570919625, 570919625, 570919625, 1496195709, 1496195709, 1496195709, 5743845611, 8271306199, 8271306199, 8271306199

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.

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

odQ[n_] := OddQ[DigitCount[n, 2, 1]]; odExp[n_] := Module[{e = 1, p = n^2}, If[! odQ[n], While[odQ[p], p *= n; e++]]; e]; seq[nmax_] := Module[{e, emax = 1, n = 3, s = {}}, Do[e = odExp[n]; If[e > emax, s = Join[s, ConstantArray[n, e - emax]]; emax = e], {n, 3, nmax, 2}]; s]; seq[10000] (* Amiram Eldar, Aug 03 2023 *)

a(26)-a(36) 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

A230495 a(n) is the minimal odd evil k, such that k^i, i=1,2,...,n, all are evil, and a(n)=0, if there is no such k.

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A230496 a(n) is the minimal odd odious k > 1, such that k^i, i=1,2,...,n, all are odious, or a(n)=0, if there is no such k.

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A230497 a(n), n>=2, is the minimal odd evil k, such that k^i, i=2,3,...,n, all are odious, 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 (2*n-1)*(2*i-1), i=1,2,...,k, which are all evil or all odious (A000069, A001969).

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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).