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A363770 Integers k such that the number of binary partitions of k is not a sum of three squares.

20, 21, 36, 37, 68, 69, 80, 81, 116, 117, 132, 133, 144, 145, 180, 181, 212, 213, 228, 229, 260, 261, 272, 273, 308, 309, 320, 321, 340, 341, 356, 357, 404, 405, 420, 421, 452, 453, 464, 465, 500, 501, 516, 517, 528, 529, 564, 565, 576, 577, 596, 597, 612, 613, 660, 661, 676, 677

Offset: 1

Maciej Ulas, Jun 21 2023

An infinite sequence.

			a(1)=20 because b(20)=60 is not a sum of three squares and for i=1, ..., 19, the numbers b(i), i=1,...,19 are sums of three squares, where b(i) is the number of binary partitions of n.

Bartosz Sobolewski and Maciej Ulas, Values of binary partition function represented by a sum of three squares, arXiv:2211.16622 [math.NT], 2023.

Cf. A004215, A018819, A363769.

bin[n_] :=
 bin[n] =
  If[n == 0, 1,
   If[Mod[n, 2] == 0, bin[n - 1] + bin[n/2],
    If[Mod[n, 2] == 1, bin[n - 1]]]];
B := {}; Do[
 If[Mod[bin[n]/4^IntegerExponent[bin[n], 4], 8] == 7,
  AppendTo[B, n]], {n, 1000}];
B

Each term is equal to 2*b(m) or 2*b(m)+1 for some m, where b(m) = A363769(m).

A363769 Integers k such that the number of binary partitions of 2k is not a sum of three squares.

Original entry on oeis.org

10, 18, 34, 40, 58, 66, 72, 90, 106, 114, 130, 136, 154, 160, 170, 178, 202, 210, 226, 232, 250, 258, 264, 282, 288, 298, 306, 330, 338, 354, 360, 378, 394, 402, 418, 424, 442, 450, 456, 474, 490, 498, 514, 520, 538, 544, 554, 562, 586, 594, 610, 616, 634, 640, 650, 658, 674, 680, 698

Offset: 1

Maciej Ulas, Jun 21 2023

An infinite sequence.

			a(1)=10 because each b(20)=60 is not a sum of three squares and for i=1, ..., 9, the numbers b(2)=2, b(4)=4, b(6)=6, b(8)=10, b(10)=14, b(12)=20, b(14)=26, b(16)=36, b(18)=46 are sums of three squares, where b(i) is the number of binary partitions of n.

Bartosz Sobolewski and Maciej Ulas, Values of binary partition function represented by a sum of three squares, arXiv:2211.16622 [math.NT], 2023.

Cf. A018819, A353219, A010060 (0 -> 1 & 1 -> -1).

bin[n_] :=
 bin[n] =
  If[n == 0, 1,
   If[Mod[n, 2] == 0, bin[n - 1] + bin[n/2],
    If[Mod[n, 2] == 1, bin[n - 1]]]];
A := {}; Do[
 If[Mod[bin[2 n]/4^IntegerExponent[bin[2 n], 4], 8] == 7,
  AppendTo[A, n]], {n, 1000}];
A

Numbers of the form {2^(2*k+1)*(8*r+2*t_{r}+3): k, r nonnegative integers} and t_{r} is r-th term of the Prouhet-Thue-Morse sequence on the alphabet {-1, +1}, i.e., t_{r} = (-1)^{s_{2}(r)}, where s_{2}(r) is the sum of binary digits of r. We have t_{r} = (-1)^A010060(r).

A082434 Let q_n be least prime > x_n := 1 + 2*n!; sequence gives a(n) = q_n-x_n+1.

Original entry on oeis.org

3, 3, 5, 5, 11, 7, 11, 11, 29, 29, 13, 71, 29, 19, 19, 37, 23, 23, 41, 29, 53, 53, 89, 29, 41, 41, 59, 227, 79, 41, 101, 73, 61, 41, 37, 59, 67, 127, 67, 223, 71, 43, 53, 71, 71, 113, 239, 71, 313, 157

Offset: 1

Maciej Ulas (maciejulas(AT)poczta.onet.pl), Apr 25 2003

			a(4) = 5 because 2*4!+1=49, the next prime is 51 and the difference between 53 and 48 is 13.

Do[If[PrimeQ[NextPrime[2*n!+1]-2*n! ], Print[n]], {n, 100}]
f[x_]:=Module[{n=2x!},NextPrime[n+1]-n]
f/@Range[60]  (* Harvey P. Dale, Feb 14 2011 *)

vector(100,n,nextprime(2*n!+2)-2*n!) \\ Charles R Greathouse IV, Feb 14 2011

a(n)=nextprime(2*n!+1)-2*n!, where nextprime is A151800.

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A363770 Integers k such that the number of binary partitions of k is not a sum of three squares.

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A363769 Integers k such that the number of binary partitions of 2k is not a sum of three squares.

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A082434 Let q_n be least prime > x_n := 1 + 2*n!; sequence gives a(n) = q_n-x_n+1.

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