A298867 -id:A298867 - cp's OEIS frontend

A298865 The primes p and products 4*p in increasing order.

2, 3, 5, 7, 8, 11, 12, 13, 17, 19, 20, 23, 28, 29, 31, 37, 41, 43, 44, 47, 52, 53, 59, 61, 67, 68, 71, 73, 76, 79, 83, 89, 92, 97, 101, 103, 107, 109, 113, 116, 124, 127, 131, 137, 139, 148, 149, 151, 157, 163, 164, 167, 172, 173, 179, 181, 188, 191, 193

Offset: 1

Clark Kimberling, Feb 13 2018

Conjecture: except for the first term, these are the nonsquares n for which there is a unique pair (x,y) such that x^2 - y^2 = n and x > y >= 0; see A257408.

Clark Kimberling, Table of n, a(n) for n = 1..1000

Cf. A000040, A298866, A298867.

z = 10000; u = Prime[Range[z]]; w = Take[Union[u, 4 u], z]; (* A298865 *)
p[n_] := If[MemberQ[u, w[[n]]], 0, 1];
t = Table[p[n], {n, 1, z}];
Flatten[Position[t, 0]];  (* A298866 *)
Flatten[Position[t, 1]];  (* A298867 *)

A298866 Positions of primes p when all p and 4*p are arranged in increasing order.

Original entry on oeis.org

1, 2, 3, 4, 6, 8, 9, 10, 12, 14, 15, 16, 17, 18, 20, 22, 23, 24, 25, 27, 28, 30, 31, 32, 34, 35, 36, 37, 38, 39, 42, 43, 44, 45, 47, 48, 49, 50, 52, 54, 55, 56, 58, 59, 60, 61, 62, 64, 65, 66, 67, 69, 70, 72, 73, 74, 76, 77, 78, 79, 80, 83, 84, 85, 86, 88

Offset: 1

Clark Kimberling, Apr 14 2018

			The joint ranking begins with 2,3,5,7,8,11,12,13,17,19,20, as in A298865, so that ranks occupied by primes are 1,2,3,4,6,7,8,10,...

Clark Kimberling, Table of n, a(n) for n = 1..1000

Cf. A000040, A298866, A298867 (complement).

z = 200; u = Prime[Range[z]]; w = Take[Union[u, 4 u], z]; (* A298865 *)
p[n_] := If[MemberQ[u, w[[n]]], 0, 1];
t = Table[p[n], {n, 1, z}];
Flatten[Position[t, 0]]  (* A298866 *)
Flatten[Position[t, 1]]  (* A298867 *)

cp's OEIS Frontend

A298865 The primes p and products 4*p in increasing order.

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