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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A302022 Primitive terms from A005279.

Original entry on oeis.org

6, 15, 20, 28, 35, 63, 77, 88, 91, 99, 104, 110, 117, 130, 143, 153, 170, 187, 190, 209, 221, 238, 247, 266, 272, 299, 304, 322, 323, 325, 357, 368, 391, 399, 425, 437, 464, 475, 483, 493, 496, 506, 513, 527, 551, 575, 589, 609, 621, 638, 651, 667, 682, 703, 713, 725, 754, 759, 775, 777, 783, 806, 814
Offset: 1

Views

Author

David A. Corneth, Mar 31 2018

Keywords

Comments

Also numbers k such that k is in A005279 but none of the proper divisors of k are.
All terms k are composites; if k is prime then it's not in A005279 hence not here. If k = m * t and t < m < 2*t then m and t are coprime. If g = gcd(t, m) > 1 then the integer k / g^2 is in A005279. If there is some term u*t where with u > 2*t and gcd(u, t) = 1 then there is some m * t' with gcd(m, t') = 1 such that m*t' | t * u and t * u wouldn't be in the sequence. if u = 2*t then gcd(u, t) = t which can't happen.
It could be that both m and t are composite, for example, t = 53^2 and m = 5^5 gives the term 53^2 * 5^5.
Interestingly, k = m * t where t < m < 2 * t and m * t is in A005279 and m, t coprime gives A106430; this sequence is a subsequence of A106430.

Examples

			77 is a term since it is in A005279 and 77 is not of the form A005279(i)*t for t > 1.

Links

Crossrefs

Subsequence of A020886 and hence of A005279.
Cf. A106430, A301989, A302296.

Programs

  • PARI
    is005279(n) = my(d=divisors(n)); for(i=3, #d, if(d[i]<2*d[i-1], return(1))); 0;
is(n) = if (is005279(n), d = divisors(n); for (k=1, #d-1, if (is005279(d[k]), return (0));); return(1);); \\ Altug Alkan, Apr 14 2018
upto(n) = {my(res = List()); for(i = 2, sqrtint(n), for(j = i+1, min(2 * i - 1, n\i), if(gcd(i, j) == 1, if(is(i*j), listput(res, i*j))))); listsort(res); return(res)} \\ David A. Corneth, Apr 15 2018

Formula

A301989(a(n)) = 1.

A302296 Positive numbers that can be written in exactly one way as i*j*k with i < j < 2*i.

Original entry on oeis.org

6, 15, 18, 20, 28, 35, 63, 75, 77, 78, 88, 91, 99, 100, 102, 104, 110, 114, 117, 130, 138, 143, 153, 170, 174, 175, 186, 187, 189, 190, 196, 209, 221, 222, 238, 245, 246, 247, 258, 266, 272, 282, 297, 299, 304, 318, 322, 323, 325, 351, 354, 357, 366, 368, 391, 399, 402, 425, 426, 429, 437, 438
Offset: 1

Views

Author

Robert Israel, Apr 04 2018

Keywords

Comments

Numbers n such that A301989(n)=1.

Examples

			a(5)=28 is in the sequence because 28 = 4*7*1 is the only way to write 28=i*j*k with i < j < 2*i.

Links

Crossrefs

Cf. A301989, A302022.

Programs

  • Maple
    N:= 1000: # to get all terms <= N
V:= Vector(N):
for i from 1 to isqrt(N-1) do
  for j from i+1 to min(floor(N/i),2*i-1) do
    for k from 1 to floor(N/(i*j)) do
      n:= i*j*k;
      V[n]:= V[n]+1;
od od od:
select(t -> V[t]=1, [$1..N]);
  • Mathematica
    M = 1000;
V = Table[0, {M}];
For[i = 1, i <= Sqrt[M-1], i++,
  For[j = i+1, j <= Min[Floor[M/i], 2i-1], j++,
    For[k = 1, k <= Floor[M/(i j)], k++,
      n = i j k;
      V[[n]] = V[[n]]+1;
]]];
Position[V, 1] // Flatten (* Jean-François Alcover, Apr 29 2019, after Robert Israel *)
  • PARI
    list(lim)=my(v=List(),u=vectorsmall(lim\=1),t); for(i=1, sqrtint(lim), for(j=i+1,min(lim\i,2*i-1), t=i*j; forstep(n=t,lim,t, u[n]++))); for(i=1,#u, if(u[i]==1, listput(v,i))); Vec(v) \\ Charles R Greathouse IV, Apr 05 2018
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