A302022 Primitive terms from A005279.

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

Author

David A. Corneth, Mar 31 2018

Keywords

• nonn

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

• Amiram Eldar, Table of n, a(n) for n = 1..10000

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.