A378294 Nonnegative norms of ideals in Q(sqrt(10), sqrt(26)).

0, 1, 2, 4, 5, 8, 9, 10, 13, 16, 18, 20, 25, 26, 32, 36, 37, 40, 45, 49, 50, 52, 64, 65, 67, 72, 74, 79, 80, 81, 83, 90, 98, 100, 104, 117, 121, 125, 128, 130, 134, 144, 148, 158, 160, 162, 163, 166, 169, 180, 185, 191, 196, 197, 199, 200, 208, 225, 227, 234, 242, 245, 250

Offset

0,3

Comments

The number of terms n up to x is asymptotic to c*x/log(x)^(3/4) for an explicitly computable constant c.

Links

Table of n, a(n) for n=0..62.

Formula

A positive integer n is in this sequence if and only if all prime factors of n congruent to {3, 7, 11, 17, 19, 21, 23, 27, 29, 31, 33, 41, 43, 47, 51, 53, 57, 59, 61, 63, 69, 71, 73, 77, 87, 89, 97, 99, 101, 103, 107, 109, 111, 113, 119, 127, 131, 133, 137, 139, 141, 147, 149, 151, 153, 157, 161, 167, 171, 173, 177, 179, 181, 183, 189, 193, 201, 207, 211, 217, 219, 223, 229, 233, 237, 239, 241, 243, 249, 251, 257, 259, 261, 263, 269, 271, 277, 279, 281, 283, 287, 291, 297, 301, 303, 309, 313, 319, 327, 331, 337, 339, 341, 343, 347, 349, 353, 359, 363, 367, 369, 371, 373, 379, 381, 383, 387, 389, 393, 401, 407, 409, 411, 413, 417, 419, 421, 423, 431, 433, 443, 447, 449, 451, 457, 459, 461, 463, 467, 469, 473, 477, 479, 487, 489, 491, 493, 497, 499, 501, 503, 509, 513, 517} mod 520 occur with even exponents.

Example

74=2*37. Using the formula above, 74 is in this sequence.
849=3*283. Using the formula above, 849 is not in this sequence, although 849=1849-1000=43^2-10*100=43^2-10*10^2 and 849==329 (mod 520).

Prog

(Magma)
IsRepresentablePrime:=func<p, D | D eq 0 select false else IsSquare(D) select true else KroneckerSymbol(FundamentalDiscriminant(D), p) in [0, 1]>;
IsRepresentableMulti:=func<p, S | forall{k: k in Subsets(S) | IsRepresentablePrime(p, &*k)}>;
IsRepresentablePoly:=func<n, S | n eq 0 or (Min(S) gt 0 or n gt 0) and forall{t: t in Factorization(n) | t[2] mod 2 eq 0 or IsRepresentableMulti(t[1], S)}>;
[n: n in [0..250] | IsRepresentablePoly(n, {10, 26})];

Crossrefs

Cf. A378295.

Sequence in context: A245226 A034026 A125022 * A362295 A069011 A353386

Adjacent sequences: A378291 A378292 A378293 * A378295 A378296 A378298

Keyword

nonn,new

Author

Jovan Radenkovicc, Nov 22 2024

Status

approved