A231978 Numbers the squares of which are in A231558.

59, 103, 193, 229, 251, 313, 431, 761, 929, 1019, 1087, 1279, 1367, 1423, 1447, 1597, 1721, 1783, 1867, 2237, 2243, 2999, 3083, 3119, 3169, 3229, 3467, 3673, 3847, 3853, 3889, 3943, 4057, 4091, 4153, 4219, 4273, 4519, 4751, 4787, 5039, 5119, 5471, 5573

Offset: 1

Author

Vladimir Shevelev and Peter J. C. Moses, Nov 16 2013

Keywords

• nonn
• base

Comments

Prime p is in the sequence, if and only if p, p^2, p^3, p^4, p^2+1 and p^2+p all are odious (A000069). We conjecture that the sequence contains also composite numbers, but the first one should be very large.

Examples

			59^2=3481 is odious together with 59, 59^3=205379, 59^4=12117361, 59^2+1=3482 and 59^2 + 59=3540. Thus 59 is in the sequence.
		

Links

• Peter J. C. Moses, Table of n, a(n) for n = 1..2500

Crossrefs

Cf. A000069, A231558.

Programs

Mathematica

odiousQ[n_]:=OddQ[DigitCount[n,2][[1]]]; selQ[n_]:=Apply[And,Map[odiousQ, Flatten[Map[{n+#,n*#,n/ #}&, Divisors[n]]]]]; Sqrt[Select[Range[3,5000]^2, (!PrimeQ[#]) && OddQ[#] && odiousQ[#] && selQ[#]&]] (* Peter J. C. Moses, Nov 16 2013 *)

Formula

A230500((a(n)+1)/2)>=4.