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.

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A387216 Numbers that have at least two prime factors (counting multiplicity) congruent to 1 mod 3.

Original entry on oeis.org

49, 91, 98, 133, 147, 169, 182, 196, 217, 245, 247, 259, 266, 273, 294, 301, 338, 343, 361, 364, 392, 399, 403, 427, 434, 441, 455, 469, 481, 490, 494, 507, 511, 518, 532, 539, 546, 553, 559, 588, 589, 602, 637, 651, 665, 676, 679, 686, 703, 721, 722, 728, 735, 741, 763, 777, 784, 793, 798
Offset: 1

Views

Author

Michel Marcus, Aug 22 2025

Keywords

Links

Crossrefs

Cf. A002476 (primes congruent to 1 mod 3), A050931 (at least one).

Programs

  • Magma
    res := [];for n in [1..1000] do L := [ f[2] : f in Factorization(n) | f[1] mod 3 eq 1 ]; count := (#L eq 0) select 0 else &+L;if count gt 1 then Append(~res, n); end if; end for; res;
 // Vincenzo Librandi, Aug 24 2025
  • Mathematica
    ff[{m_,n_}]:=Table[m,n];Select[Range[798],Count[Mod[ff/@FactorInteger[#]//Flatten,3],1]>1&] (* James C. McMahon, Aug 22 2025 *)
  • PARI
    isok(k) = my(f=factor(k)); sum(i=1, #f~, if ((f[i,1]%3) == 1, f[i,2])) >= 2;

A300303 Squares that are not of the form x^2 + x*y + y^2, where x and y are positive integers.

Original entry on oeis.org

0, 1, 4, 9, 16, 25, 36, 64, 81, 100, 121, 144, 225, 256, 289, 324, 400, 484, 529, 576, 625, 729, 841, 900, 1024, 1089, 1156, 1296, 1600, 1681, 1936, 2025, 2116, 2209, 2304, 2500, 2601, 2809, 2916, 3025, 3364, 3481, 3600, 4096, 4356, 4624, 4761, 5041, 5184, 5625, 6400, 6561, 6724, 6889, 7225, 7569
Offset: 1

Views

Author

Altug Alkan, Mar 02 2018

Keywords

Comments

Or Loeschian numbers (A003136) that are not in A024614.
Squares that are not in this sequence are 49, 169, 196, 361, 441, 676, ...
This is the list of squares not of the form A050931(k)^2. A number n is in this sequence iff n = m^2 with m having no prime factor == 1 (mod 6). - M. F. Hasler, Mar 04 2018

Examples

			Loeschian number 25 = 5^2 is a term because 25 = x^2 + x*y + y^2 has no solution for positive integers x, y.

Crossrefs

Cf. A000548, A003136, A024614, A050931, A230780.

Programs

  • Maple
    isA024614:= proc(n) local x,y;
for x from 1 to floor(sqrt(n-1)) do
   if issqr(4*n-3*x^2) then return true fi
od:
false
end proc:
isA024614(0):= false:
remove(isA024614, [seq(i^2,i=0..200)]); # Robert Israel, Mar 02 2018
  • Mathematica
    sol[s_] := Solve[0 < x <= y && s == x^2 + x y + y^2, {x, y}, Integers];
Select[Range[0, 100]^2, sol[#] == {}&] (* Jean-François Alcover, Oct 26 2020 *)
  • PARI
    is(n,m)=issquare(n,m)&&!setsearch(Set(factor(m)[,1]%6),1) \\ second part is equivalent to is_A230780(m), this is sufficient to test (e.g., to produce a list) if we know that n = m^2. - M. F. Hasler, Mar 04 2018

Formula

a(n) = A230780(n-1)^2 for n > 1.

A374365 Numbers m such that abs(Sum_{k=1..m} [k|m]*A008683(k)*(-1)^(2*k/5)) = 0.

Original entry on oeis.org

11, 22, 31, 33, 41, 44, 55, 61, 62, 66, 71, 77, 82, 88, 93, 99, 101, 110, 121, 122, 123, 124, 131, 132, 142, 143, 151, 154, 155, 164, 165, 176, 181, 183, 186, 187, 191, 198, 202, 205, 209, 211, 213, 217, 220, 231, 241, 242, 244, 246, 248, 251, 253, 262, 264, 271
Offset: 1

Views

Author

Mats Granvik, Jul 06 2024

Keywords

Comments

Conjecture: Numbers having a prime factor congruent to 1 mod 10.

Crossrefs

Cf. A008683, A009003, A050931, A228541.

Programs

  • Mathematica
    nn = 274; Flatten[Position[ParallelTable[Abs[Sum[If[Mod[n, k] == 0, 1, 0]*((-1)^( 2*k/5))*MoebiusMu[k], {k, 1, n}]], {n, 1, nn}], 0]]
Previous Showing 11-13 of 13 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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