cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

User: Jovan Radenkovicc

Jovan Radenkovicc's wiki page.

Jovan Radenkovicc has authored 3 sequences.

A378295 Prime norms of ideals in Q(sqrt(10), sqrt(26)).

Original entry on oeis.org

2, 5, 13, 37, 67, 79, 83, 163, 191, 197, 199, 227, 293, 307, 311, 317, 397, 439, 521, 557, 569, 587, 599, 601, 613, 641, 643, 683, 719, 733, 751, 773, 787, 809, 827, 853, 877, 881, 911, 919, 947, 991, 1031, 1039, 1049, 1123, 1163, 1231, 1237, 1249, 1307, 1361, 1373, 1439, 1481, 1493
Offset: 1

Views

Author

Jovan Radenkovicc, Nov 22 2024

Keywords

Comments

Except for 2, 5 and 13, primes congruent to 1, 9, 37, 49, 67, 79, 81, 83, 93, 121, 123, 129, 159, 163, 187, 191, 197, 199, 203, 209, 213, 227, 231, 253, 267, 289, 293, 307, 311, 317, 321, 323, 329, 333, 357, 361, 391, 397, 399, 427, 437, 439, 441, 453, 471, 483, 511, 519 mod 520.
Primes in A378294.
Every prime p occurs in exactly one or all three of the sequences A038879, A038899 and A038945. This sequence lists the primes appearing in all three sequences.

Crossrefs

Cf. A038879, A038899, A038945, A378294.

Programs

  • Magma
    [p: p in PrimesUpTo(1500) | p in {2, 5, 13} or p mod 520 in [1, 9, 37, 49, 67, 79, 81, 83, 93, 121, 123, 129, 159, 163, 187, 191, 197, 199, 203, 209, 213, 227, 231, 253, 267, 289, 293, 307, 311, 317, 321, 323, 329, 333, 357, 361, 391, 397, 399, 427, 437, 439, 441, 453, 471, 483, 511, 519]];

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

Original entry on oeis.org

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

Views

Author

Jovan Radenkovicc, Nov 22 2024

Keywords

Comments

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

Examples

			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).

Crossrefs

Cf. A378295.

Programs

  • Magma
    IsRepresentablePrime:=func;
IsRepresentableMulti:=func;
IsRepresentablePoly:=func;
[n: n in [0..250] | IsRepresentablePoly(n,{10, 26})];

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.

A378048 Numbers k such that k and k^2 together use at most 4 distinct decimal digits.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 19, 20, 21, 22, 23, 25, 26, 27, 28, 30, 31, 35, 38, 40, 41, 45, 46, 50, 55, 56, 60, 63, 64, 65, 66, 68, 70, 74, 75, 76, 77, 80, 81, 83, 85, 88, 90, 91, 95, 96, 97, 99, 100, 101, 102, 105, 109, 110
Offset: 1

Views

Author

Jovan Radenkovicc, Nov 15 2024

Keywords

Comments

Problem: Is there a real constant c such that a(n) < n^c for all positive integers n?
All of A136808, A136809, A136816, ..., A137079 are subsequences. Many if not most terms of A058411, A058413, ... ("tridigital solutions") are also in this sequence; see also Hisanori Mishima's web page for some nontrivial solutions. - M. F. Hasler, Feb 02 2025

Examples

			816 is in the sequence since 816^2 = 665856 and both together use at most 4 distinct digits.
149 is not in the sequence since 149^2 = 22201 and both together use 5 distinct digits.

Links

Crossrefs

Supersequence of A136808, A136809, A136816, A136822.
Cf. A043537, A053061, A055436, A154155.

Programs

  • Magma
    [n: n in [0..1000000] | #Set(Intseq(n)) le 4 and #Set(Intseq(n) cat Intseq(n^2)) le 4];
  • Mathematica
    Select[Range[0, 110], Length[Union @@ IntegerDigits@ {#, #^2}] < 5 &] (* Amiram Eldar, Nov 15 2024 *)
  • PARI
    isok(k) = #Set(concat(digits(k), digits(k^2))) <= 4; \\ Michel Marcus, Nov 15 2024
  • PARI
    is(n)=my(s=Set(digits(n))); #s<5 && #setunion(Set(digits(n^2)),s)<5 \\ Charles R Greathouse IV, Jan 30 2025
  • PARI
    is1(n)=#setunion(Set(digits(n^2)),Set(digits(n)))<5
ok(m)=my(d=concat(apply(k->digits(lift(k)), [m,m^2]))
test(d)=my(v=List(),D=10^d); for(n=0,D-1, if(ok(Mod(n,D)), listput(v,n))); Vec(v)
res=test(8); \\ build a list of residues mod 10^8
D=diff(concat(res,res[1]+10^8)); #D
u=List(); for(n=0,10^7, if(is1(n) && !setsearch(n,res), listput(u,n))); \\ build exceptions
setminus(select(is1,[0..n]),list(n))
list(lim)=my(v=List(u)); forstep(n=0,lim,D, if(is1(n), listput(v,n))); Vec(v) \\ Charles R Greathouse IV, Jan 30 2025
  • Python
    def ok(n): return len(set(str(n)+str(n**2))) <= 4
print([k for k in range(111) if ok(k)]) # Michael S. Branicky, Nov 18 2024

Formula

A043537(A053061(a(n))) <= 4.
Trivially, a(n) >> n^1.66... where the exponent is log(10)/log(4) (A154155). - Charles R Greathouse IV, Jan 30 2025

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