A160216 -id:A160216 - cp's OEIS frontend

A160227 Union of A160216 and squares of odd terms of A160215.

3, 7, 11, 19, 23, 25, 31, 41, 43, 47, 59, 67, 71, 73, 79, 83, 89, 97, 103, 107, 127, 131, 137, 139, 151, 163, 167, 169, 179

Offset: 1

Vladimir Shevelev, May 04 2009, May 06 2009

nonn

The sequence is obtained by an application of an Eratosthenes-like sieve to A160217.

Cf. A160215 A160216 A160217 A000040

Edited by N. J. A. Sloane, May 08 2009

A160217 Minimal increasing sequence with a(1)=3 and the property that a(n) and n are both in or both not in A003159.

Original entry on oeis.org

3, 6, 7, 9, 11, 14, 15, 18, 19, 22, 23, 25, 27, 30, 31, 33, 35, 38, 39, 41, 43, 46, 47, 50, 51, 54, 55, 57, 59, 62, 63, 66, 67, 70, 71, 73, 75, 78, 79, 82, 83, 86, 87, 89, 91, 94, 95, 97, 99, 102, 103, 105, 107, 110, 111, 114, 115, 118, 119, 121, 123, 126, 127, 129, 131, 134

Offset: 1

Vladimir Shevelev, May 04 2009

nonn

The primes in this sequence give A160216.

Conjecture: Let m>3 belong to A003159. Define the sequence b(n) to be the minimal increasing sequence with b(1)=m and the property that b(n) and n are both in or both not in A003159. Then a(n)=b(n) for all n larger than some m-dependent minimum index.

			n=2 is not in A003159. So a(2) is the smallest number larger than a(1)=3 which is not in A003159. This excludes 4 and 5 which are in A003159 and leads to a(2)=6.

V. Shevelev, Several results on sequences which are similar to the positive integers, arXiv:0904.2101 [math.NT], 2009.

Cf. A003159, A007814, A010060, A160216, A159619.

Cf. A004760, A160230. [Vladimir Shevelev, May 05 2009]

a35263[n_] := 1 - Mod[IntegerExponent[n, 2], 2];
a[1] = 3; a[n_] := a[n] = For[k = a[n - 1] + 1, True, k++, If[a35263[k] == a35263[n], Return[k]]];
Array[a, 66] (* Jean-François Alcover, Jul 28 2018 *)

is(n) = valuation(n, 2)%2==0; \\ A003159
nexta(a, n) = {my(k=a+1, isn = is(n)); while (is(k) != isn, k++); k;};
lista(nn) = {my(a = 3); print1(a, ", "); for (n=2, nn, a = nexta(a, n); print1(a, ", "););} \\ Michel Marcus, Dec 15 2018

a(n+1) = min{ m>a(n): A035263(m)=A035263(n+1) }.
a(n)=2n+1, if A007814(n) is even. a(n)=2n+2, if A007814(n) is odd.
A010060(a(n))=1-A010060(n)
For n>=1, A010060(a(n))= A010060(A004760(n+1)). See also A160230. [Vladimir Shevelev, May 05 2009]

Edited by R. J. Mathar, May 08 2009

A160215 Primes congruent to 2^k+1 (mod 2^(k+1)), where k is any even integer >=0.

Original entry on oeis.org

2, 5, 13, 17, 29, 37, 53, 61, 101, 109, 113, 149, 157, 173, 181, 193, 197, 229, 241, 257, 269, 277, 293, 317, 337, 349, 373, 389, 397, 401, 421, 433, 449, 461, 509, 541, 557, 577, 593, 613, 653, 661, 677, 701, 709, 733, 757, 769, 773, 797, 821, 829, 853, 877

Offset: 1

Vladimir Shevelev, May 04 2009

nonn

If A(x) is the counting function of the terms not exceeding x, then A(x) grows similarly to Pi(x)/3, see A000720.

Lim_{x -> inf.} the number of terms < x in A160216/A160215 => 2. - Robert G. Wilson v, May 31 2009

Vincenzo Librandi, Table of n, a(n) for n = 1..2000

Cf. A000040.

fQ[n_] := Mod[ Flatten[ FactorInteger[n - 1]] [[2]], 2] == 0; Select[ Prime@ Range@ 155, fQ@# &] (* Robert G. Wilson v, May 31 2009 *)

A000040 \ A160216.

{prime(k) : A023506(k) is even}. - R. J. Mathar, May 08 2009

Edited by R. J. Mathar, May 08 2009

More terms from Robert G. Wilson v, May 31 2009

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A160227 Union of A160216 and squares of odd terms of A160215.

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A160217 Minimal increasing sequence with a(1)=3 and the property that a(n) and n are both in or both not in A003159.

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A160215 Primes congruent to 2^k+1 (mod 2^(k+1)), where k is any even integer >=0.

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