A000548 -id:A000548 - cp's OEIS frontend

A068952 Squares in A068949.

1, 4, 9, 16, 36, 49, 64, 81, 100, 121, 144, 196

Offset: 1

Francois Jooste (phukraut(AT)hotmail.com), Mar 15 2002

This sequence, so far, looks like A000548, except with the addition of 100. Is there a relationship?

			a(4)=16 since the fourth square in A068949 is 16.

Cf. A068809, A068949, A000548.

a(12) from Giovanni Resta, Jun 27 2018

A143575 Numbers m such that A143574(m) = m.

Original entry on oeis.org

0, 1, 4, 5, 9, 10, 13, 16, 17, 20, 26, 29, 34, 36, 37, 40, 41, 45, 49, 52, 53, 58, 61, 64, 68, 73, 74, 80, 81, 82, 89, 90, 97, 101, 104, 106, 109, 113, 116, 117, 121, 122, 136, 137, 144, 146, 148, 149, 153, 157, 160, 164, 173, 178, 180, 181, 193, 194, 196, 197, 202

Offset: 1

Reinhard Zumkeller, Aug 24 2008

nonn

A000161(a(n)) = 1.

R. Zumkeller, Table of n, a(n) for n = 1..10000

A000548 is a subsequence.

A235142 Numbers k such that A235141(k) = -1.

Original entry on oeis.org

4, 10, 22, 32, 64, 84, 108, 132, 186, 214, 284, 360, 446, 490, 590, 642, 694, 746, 930, 990, 1192, 1266, 1342, 1568, 1738, 2086, 2180, 2276, 2470, 2572, 2668, 2780, 3326, 3556, 3680, 3922, 4298, 4430, 4560, 4832, 4968

Offset: 1

Rajan Murthy, Jan 03 2014

nonn

The positions reflect square radii which are uniquely twice a square integer, that is, the completion of only one square on the y = x line.

			For n = 2, a(2) = 10.  The 10th term of A235141 is -1 corresponding to the square radius of an origin centered circle increasing from the open interval(5,8) to exactly 8.

A000548(n) = (A001481(1 + a(n)/2 ) )/2.

Cf. A235141, A234300.

a(n) = A235386(n+1) - 1.

A235386 Numbers k such that A235141(k) = 1.

Original entry on oeis.org

1, 5, 11, 23, 33, 65, 85, 109, 133, 187, 215, 285, 361, 447, 491, 591, 643, 695, 747, 931, 991, 1193, 1267, 1343, 1569, 1739, 2087, 2181, 2277, 2471, 2573, 2669, 2781, 3327, 3557, 3681, 3923, 4299, 4431, 4561, 4833, 4969

Offset: 1

Rajan Murthy, Jan 08 2014

nonn

The positions reflect square radii which are uniquely twice a square integer, that is, the inclusion of only one square on the y = x line.

			for n=3, a(3) = 11. The eleventh term of A235141 is 1 reflecting an increase in the square radius of the circle from exactly 8 to the open interval of (8,9).

A000548(n) = A001481(1 + (a(n+1)-1)/2)/2.

Cf. A235141, A234300.

a(n+1) = A235142(n) + 1, a(1)=1.

A300303 Squares that are not of the form x^2 + x*y + y^2, where x and y are positive integers.

Original entry on oeis.org

0, 1, 4, 9, 16, 25, 36, 64, 81, 100, 121, 144, 225, 256, 289, 324, 400, 484, 529, 576, 625, 729, 841, 900, 1024, 1089, 1156, 1296, 1600, 1681, 1936, 2025, 2116, 2209, 2304, 2500, 2601, 2809, 2916, 3025, 3364, 3481, 3600, 4096, 4356, 4624, 4761, 5041, 5184, 5625, 6400, 6561, 6724, 6889, 7225, 7569

Offset: 1

Altug Alkan, Mar 02 2018

Or Loeschian numbers (A003136) that are not in A024614.
Squares that are not in this sequence are 49, 169, 196, 361, 441, 676, ...
This is the list of squares not of the form A050931(k)^2. A number n is in this sequence iff n = m^2 with m having no prime factor == 1 (mod 6). - M. F. Hasler, Mar 04 2018

			Loeschian number 25 = 5^2 is a term because 25 = x^2 + x*y + y^2 has no solution for positive integers x, y.

Cf. A000548, A003136, A024614, A050931, A230780.

isA024614:= proc(n) local x,y;
for x from 1 to floor(sqrt(n-1)) do
   if issqr(4*n-3*x^2) then return true fi
od:
false
end proc:
isA024614(0):= false:
remove(isA024614, [seq(i^2,i=0..200)]); # Robert Israel, Mar 02 2018

sol[s_] := Solve[0 < x <= y && s == x^2 + x y + y^2, {x, y}, Integers];
Select[Range[0, 100]^2, sol[#] == {}&] (* Jean-François Alcover, Oct 26 2020 *)

is(n,m)=issquare(n,m)&&!setsearch(Set(factor(m)[,1]%6),1) \\ second part is equivalent to is_A230780(m), this is sufficient to test (e.g., to produce a list) if we know that n = m^2. - M. F. Hasler, Mar 04 2018

a(n) = A230780(n-1)^2 for n > 1.

cp's OEIS Frontend

A068952 Squares in A068949.

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A143575 Numbers m such that A143574(m) = m.

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A235142 Numbers k such that A235141(k) = -1.

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A235386 Numbers k such that A235141(k) = 1.

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A300303 Squares that are not of the form x^2 + x*y + y^2, where x and y are positive integers.

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