

A220221


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


1



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
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OFFSET

1,2


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) 131142.
Index entries for linear recurrences with constant coefficients, signature (3,2).


FORMULA

a(n) = 3*a(n1)2*a(n2) for n>9.  Colin Barker, Nov 06 2014
G.f.: x*(12*x^82*x^710*x^6+4*x^52*x^42*x^32*x^2+1) / ((x1)*(2*x1)).  Colin Barker, Nov 06 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 *)


PROG

(PARI) is(n)=n%2 && hammingweight(n^2)<4 \\ Charles R Greathouse IV, Dec 10 2012
(PARI) Vec(x*(12*x^82*x^710*x^6+4*x^52*x^42*x^32*x^2+1)/((x1)*(2*x1)) + O(x^100)) \\ Colin Barker, Nov 06 2014
(MAGMA) I:=[1, 3, 5, 7, 9, 17, 23, 33, 65, 129]; [n le 10 select I[n] else 3*Self(n1)2*Self(n2): n in [1..50]]; // Vincenzo Librandi, Nov 07 2014


CROSSREFS

Cf. A212191 (exactly 3 powers).
Sequence in context: A084229 A191356 A144753 * A212292 A270837 A057482
Adjacent sequences: A220218 A220219 A220220 * A220222 A220223 A220224


KEYWORD

nonn,base,easy


AUTHOR

John W. Layman, Dec 07 2012


EXTENSIONS

Extended by T. D. Noe, Dec 07 2012


STATUS

approved



