cp's OEIS Frontend

Conjecture: a(n) > 0 for all n > 1; in other words, for any integer n > 1 there is a nonnegative integer k such that either n^2 - 2*4^k or n^2 - 5*4^k can be written as the sum of two squares. Moreover, a(n) = 1 only for n = 2^k with k > 0.

This conjecture is stronger than the first conjecture in A300448. We have verified that a(n) > 0 for all n = 2..5*10^7.

Consider positive integers c not divisible by 4 such that for any integer n > 1 there is a nonnegative integer k for which n^2 - 2*4^k or n^2 - c*4^k can be written as the sum of two squares. Our computation for n up to 3*10^7 shows that the only candidates for values of c smaller than 160 are 5, 17, 18, 26, 29, 41, 45, 65, 74, 89, 98, 101, 113, 122, 125, 146, 149, 153. These numbers have the form 9^a*(3*b+2) with a and b nonnegative integers and the p-adic order of 3*b+2 is even for any prime p == 3 (mod 4). For n = 42211965 there is no nonnegative integer k such that n^2 - 2*4^k or n^2 - 162*4^k can be written as the sum of two squares.

Qing-Hu Hou at Tianjin Univ. reported that he had verified a(n) > 0 for n up to 10^9. - Zhi-Wei Sun, Mar 14 2018

Qing-Hu Hou found that 29, 65, 113 should be excluded from the candidates. In fact, for c = 29, 65, 113 there is no nonnegative integer k such that N(c)^2 - 2*4^k or N(c)^2 - c*4^k can be written as the sum of two squares, where N(29) = 51883659, N(65) = 56173837 and N(113) = 65525725. - Zhi-Wei Sun, Mar 23 2018

a(n) > 0 for 1 < n < 6*10^9. - Giovanni Resta, Jun 14 2019

Clearly, a(2*n) > 0 if a(n) > 0. We note that 52603423 is the first value of n > 1 with a(n) = 0.

See also A302982 and A302983 for related things.

Conjecture: a(n) > 0 for all n > 3.

Clearly, a(2*n) > 0 if a(n) > 0. We have verified a(n) > 0 for all n = 4..6*10^9.

See also A302982 and A302984 for similar conjectures.

Conjecture: a(n) > 0 for all n > 1.

We call this the 2-3 conjecture. It is simialr to the author's 2-5 conjecture which states that A300510(n) > 0 for all n > 1.

We have verified that a(n) > 0 for all n = 2..5*10^7.

It is known that the number of ways to write a positive integer n as x^2 + 2*y^2 with x and y integers is twice the difference |{d > 0: d|n and d == 1,3 (mod 8)}| - |{d>0: d|n and d == 5,7 (mod 8)}|.

Conjecture: a(n) > 0 for all n > 3.

This is equivalent to the author's conjecture in A302983. We have verified a(n) > 0 for all n = 4...6*10^9.

See also A302982 and A302984 for similar conjectures.

Conjecture 1 (1-2-3 Conjecture): a(n) > 0 for all n >= 0. In other words, any positive odd integer m can be written as x^2 + y^2 + z^2 + w^2 with x, y, z, w nonnegative integers such that x + 2*y + 3*z = 2^k for some positive integer k.

Conjecture 2 (Strong Version of the 1-2-3 Conjecture): For any integer m > 4627 not congruent to 0 or 2 modulo 8, we can write m as x^2 + y^2 + z^2 + w^2 with x, y, z, w nonnegative integers such that x + 2*y + 3*z = 4^k for some positive integer k.

We have verified Conjectures 1 and 2 for m up to 5*10^6. Conjecture 2 implies that A299924(n) > 0 for all n > 0.

By Theorem 1.2(v) of the author's 2017 JNT paper, any positive integer n can be written as x^2 + y^2 + z^2 + 4^k with k, x, y, z nonnegative integers.

See also A338094 and A338095 for similar conjectures.

Conjecture: a(n) > 0 for all n > 1, and a(n) = 1 only for n = 5, 13, 25, 29, 59, 61, 79, 91, 95, 101, 103, 1315, 2^k (k = 1,2,3,...), 2^(2k+1)*m (k = 0,1,2,... and m = 3, 5, 7, 11, 15, 19, 23).

Note the difference between the current sequence and A300356.

In the comments of A300219, the author conjectured that a positive square n^2 can be written as x^2 + y^2 + z^2 + w^2 with x,y,z,w nonnegative integers such that both x and x + 3*y are powers of 4 unless n has the form 4^k*81503 with k a nonnegative integer. Since 81503^2 = 208^2 + 16^2 + 51167^2 + 63440^2 with 16 = 4^2 and 208 + 3*16 = 4^4, this implies that any positive square can be written as x^2 + y^2 + z^2 + w^2 with x,y,z,w nonnegative integers such that x or y is a power of 4 and x + 3y is also a power of 4. We also conjecture that for any positive integer n not of the form 4^k*m (k =0,1,... and m = 2, 7) we can write n^2 as x^2 + y^2 + z^2 + w^2 with x,y,z,w nonnegative integers such that x or y is a power of 4 and x + 2*y is also a power of 4.

The Square Conjecture in A301471 implies that a(n) >= 0 for all n > 1.

It is known that a positive integer n has the form x^2 + 2*y^2 with x and y integers if and only if the p-adic order of n is even for any prime p == 5 or 7 (mod 8).

Numbers t such that a(t) = 0 are 2, 3, 5, 6, 10, 11, 13, 14, 18, 19, 21, 26, 27, 29, 34, 37, ... - Altug Alkan, Mar 26 2018

Conjecture: a(n) > 0.6*log_2(log_2 n) for all n > 2, and also lim inf_{n->infinity} a(n)/(log n) = 0.

The author's Square Conjecture in A301471 would imply that a(n) >= 0 for all n > 1. We have verified that a(n) > 0.6*log_2(log_2 n) for all n = 3..4*10^9. For n = 2857932461, we have a(n) = 3 and 0.603 < a(n)/log_2(log_2 n) < 0.604.

It is known that a positive integer n has the form x^2 + 2*y^2 with x and y integers if and only if the p-adic order of n is even for any prime p == 5 or 7 (mod 8).

Conjecture: a(n) > 0 for all n > 1.

Note that a nonnegative integer m is the sum of two triangular numbers if and only if 4*m+1 (or 8*m+2) can be written as the sum of two squares.

We have verified a(n) > 0 for all n = 2..4*10^8. See also the related sequences A303233 and A303234.