A256944 -id:A256944 - cp's OEIS frontend

A257282 Numbers whose square is not the sum of two consecutive nonsquares.

0, 1, 2, 3, 4, 6, 7, 8, 10, 12, 14, 16, 17, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 41, 42, 44, 46, 48, 50, 52, 54, 56, 58, 60, 62, 64, 66, 68, 70, 72, 74, 76, 78, 80, 82, 84, 86, 88, 90, 92, 94, 96, 98, 99, 100, 102, 104, 106, 108, 110, 112, 114, 116

Offset: 1

M. F. Hasler, May 08 2015

See A256944 for further information.

Union of even integers and A001333. - Ivan Neretin, Aug 04 2016

			5 is not in the sequence because 5^2 = 25 = 12 + 13 is the sum of two consecutive numbers both of which are not squares.
All even numbers are in the sequence because (2k)^2 = 4k^2 cannot be written as sum of two consecutive numbers and 2k^2 is not a square.
An odd number n is in the sequence if one of the two numbers next to n^2/2 is a square.

Ivan Neretin, Table of n, a(n) for n = 1..10001

Cf. A256944.

Union[#, Range[0, Max@ #, 2]] &@ Numerator[Convergents[Sqrt@ 2, 7]] (* Michael De Vlieger, Aug 06 2016, after Harvey P. Dale at A001333 *)

is(n)={n%2==0 || issquare(n^2\2) || issquare(n^2\2+1)}

a(n) = sqrt(A256944).

a(n) ~ 2n. [Following Charles R Greathouse IV's formula for A256944.]

A257292 Numbers whose square can be written as the sum of two consecutive nonsquares.

Original entry on oeis.org

5, 9, 11, 13, 15, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 43, 45, 47, 49, 51, 53, 55, 57, 59, 61, 63, 65, 67, 69, 71, 73, 75, 77, 79, 81, 83, 85, 87, 89, 91, 93, 95, 97, 101, 103, 105, 107, 109, 111, 113, 115, 117, 119, 121, 123, 125, 127, 129, 131

Offset: 1

M. F. Hasler, May 08 2015

nonn

Equivalently, odd numbers such that neither of the two integers next to n^2/2 is a square.
Complement of A257282 = square roots of A256944.
The odd numbers missing here are 1, 3, 7, 17, 41, 99, ... = A078057 (see also A001333 = abs(A123335)).

			9 is a term because 9^2 = 81 = 40 + 41, neither of which are square.

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A001333, A078057, A123335, A256944, A257282.

Select[Range[1, 131, 2], AllTrue[{Floor[#^2/2], Ceiling[#^2/2]}, ! IntegerQ@ Sqrt@ # &] &] (* Michael De Vlieger, Dec 11 2015 *)

select( is(n)={bittest(n,0) && !issquare(n^2\2) && !issquare(n^2\/2)}, [0..140]) \\ Corrected Jul 06 2021, thanks to an observation by Bill McEachen

A257553 Primes whose squares are not the sums of two consecutive nonsquares.

Original entry on oeis.org

2, 3, 7, 17, 41, 239, 577, 665857, 9369319, 63018038201, 489133282872437279, 19175002942688032928599, 123426017006182806728593424683999798008235734137469123231828679

Offset: 1

Juri-Stepan Gerasimov, Apr 29 2015

nonn

Primes of the form A257282(k).
2 is in this sequence, and an odd prime p is in the sequence iff either (p^2 - 1)/2 or (p^2 + 1)/2 is a square. - Wolfdieter Lang, May 07 2015
According to the Neretin comment in A257282, and as the primes of A001333 are in A086395, this is (apart from the 2) the same as A086395. - R. J. Mathar, Jan 31 2024

			2 is in the sequence because it is prime and its square 4 is in A256944: 4 is not a sum of consecutive numbers.
3 is in the sequence because it is prime and its square 9 is in A256944: 9 = 2^2 + 5.
7 is in the sequence because it is prime and its square 49 is in A256944: 49 = 24 + 5^2.
5 is not in the sequence because neither 12 nor 13 is a square.

Cf. A000290, A001601, A088165, A256944.

lim = 1000000; s = Plus @@@ (Partition[#, 2, 1] & @ Complement[Range@ lim, Range[Floor@ Sqrt[lim]]^2]); Select[Sqrt[#] & /@ Select[Range@ Floor[Sqrt[lim]]^2, ! MemberQ[s, #] &] , PrimeQ] (* Michael De Vlieger, Apr 29 2015 *)

Name clarified by Michael De Vlieger and Jon E. Schoenfield, May 03 2015

Edited by Wolfdieter Lang, May 07 2015

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A257282 Numbers whose square is not the sum of two consecutive nonsquares.

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A257292 Numbers whose square can be written as the sum of two consecutive nonsquares.

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A257553 Primes whose squares are not the sums of two consecutive nonsquares.

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