A220221 Odd positive integers k such that k^2 has at most three nonzero binary digits.

1, 3, 5, 7, 9, 17, 23, 33, 65, 129, 257, 513, 1025, 2049, 4097, 8193, 16385, 32769, 65537, 131073, 262145, 524289, 1048577, 2097153, 4194305, 8388609, 16777217, 33554433, 67108865, 134217729, 268435457, 536870913, 1073741825, 2147483649, 4294967297

Offset: 1

Author

John W. Layman, Dec 07 2012

Keywords

• nonn
• base
• easy

Comments

It is shown in the Szalay reference that if y is a term of this sequence then y=7, y=23, or y=2^t+1 for some positive t. Also see the Bennett reference.

Links

• Colin Barker, Table of n, a(n) for n = 1..1000
• Michael A. Bennett, Perfect powers with few ternary digits, INTEGERS 12A (2012), #A3.
• László Szalay, The equations 2^n+-2^m+-2^l=z^2, Indag. Math. 13 (2002) 131-142.
• Index entries for linear recurrences with constant coefficients, signature (3,-2).

Crossrefs

Cf. A212191 (exactly 3 powers).

Programs

Magma

I:=[1,3,5,7,9,17,23,33,65,129]; [n le 10 select I[n] else 3*Self(n-1)-2*Self(n-2): n in [1..50]]; // Vincenzo Librandi, Nov 07 2014

Mathematica

Select[Range[1, 1000000, 2], Total[IntegerDigits[#^2, 2]] <= 3 &] (* T. D. Noe, Dec 07 2012 *)
CoefficientList[Series[(12 x^8 - 2 x^7 - 10 x^6 + 4 x^5 - 2 x^4 - 2 x^3 - 2 x^2 + 1) / ((x - 1) (2 x - 1)), {x, 0, 50}], x] (* Vincenzo Librandi, Nov 07 2014 *)

PARI

is(n)=n%2 && hammingweight(n^2)<4 \\ Charles R Greathouse IV, Dec 10 2012

PARI

Vec(x*(12*x^8-2*x^7-10*x^6+4*x^5-2*x^4-2*x^3-2*x^2+1)/((x-1)*(2*x-1)) + O(x^100)) \\ Colin Barker, Nov 06 2014

Formula

a(n) = 3*a(n-1)-2*a(n-2) for n>9. - Colin Barker, Nov 06 2014

G.f.: x*(12*x^8-2*x^7-10*x^6+4*x^5-2*x^4-2*x^3-2*x^2+1) / ((x-1)*(2*x-1)). - Colin Barker, Nov 06 2014

Extensions

Extended by T. D. Noe, Dec 07 2012