This site is supported by donations to The OEIS Foundation. Please make a donation to keep the OEIS running. We are now in our 55th year. In the past year we added 12000 new sequences and reached 8000 citations (which often say "discovered thanks to the OEIS"). We need to raise money to hire someone to manage submissions, which would reduce the load on our editors and speed up editing. Other ways to donate

 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A307706 Number of unitary divisors of n that are smaller than sqrt((sqrt(2) - 1)*n). 0
 0, 0, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 1, 1, 1, 3, 1, 1, 2, 2, 1, 1, 1, 2, 2, 1, 1, 3, 1, 2, 1, 2, 1, 2, 1, 2, 2, 2, 1, 2, 1, 1, 2, 2, 1, 3, 1, 2, 1, 1, 2, 3, 1, 2, 2, 3, 1, 1, 1, 2, 2, 2, 1, 3, 1, 2, 1, 2, 1, 3, 2, 2, 2 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,10 COMMENTS Related to A024359: (Start) Note that all the primitive Pythagorean triangles are given by A = min{2*u*v, u^2 - v^2}, B = max{2*u*v, u^2 - v^2}, C = u^2 + v^2, where u, v are coprime positive integers, u > v and u - v is odd. As a result: (a) if n is odd, then A024359(n) is the number of representations of n to the form n = u^2 - v^2, where u, v are coprime positive integers (note that this guarantees that u - v is odd), u > v and u^2 - v^2 < 2*u*v. Let s = u + v, t = u - v, then n = s*t, where s and t are unitary divisors of n, s > t and s*t < (s^2 - t^2)/2, so t is in the range (0, sqrt((sqrt(2) - 1)*n)); (b) if n is divisible by 4, then A024359(n) is the number of representations of n to the form n = 2*u*v, where u, v are coprime positive integers (note that this also guarantees that u - v is odd because n/2 is even), u > v and 2*u*v < u^2 - v^2. So u and v must be unitary divisors of n/2, and v is in the range (0, sqrt((sqrt(2) - 1)*(n/2))). (c) if n == 2 (mod 4), then n/2 is odd, so n = 2*u*v implies that u and v are both odd, which is not acceptable, so A024359(n) = 0. Similarly, let b(n) be the number of unitary divisors of n in the range (sqrt((sqrt(2) - 1)*n), sqrt(n)) (= A034444(n)/2 - a(n) for n > 1), then the number of times B takes value n is b(n) for odd n > 1, b(n/2) if n is divisible by 4 and 0 if n = 1 or n == 2 (mod 4). (End) For k >= 2, the earliest occurence of k is at n = A132404(k)/2 if A132404(k) is even (and thus being a multiple of 4). Conjecture: this is always the case. LINKS FORMULA A024359(n) = a(n) for odd n; A024359(n) = a(n/2) for n divisible by 4. EXAMPLE The unitary divisors of 210 that are smaller than sqrt((sqrt(2) - 1)*210) = 9.3265... are 1, 2, 3, 5, 6 and 7, so a(210) = 6. Correspondingly, A024359(420) = 6. PROG (PARI) a(n) = my(i=0); for(k=1, sqrt((sqrt(2)-1)*n), if(!(n%k) && gcd(k, n/k)==1, i++)); i CROSSREFS Cf. A024359, A034444, A132404. Sequence in context: A129252 A327936 A022929 * A161102 A276329 A161101 Adjacent sequences:  A307703 A307704 A307705 * A307707 A307708 A307709 KEYWORD nonn AUTHOR Jianing Song, Apr 23 2019 STATUS approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified December 13 09:57 EST 2019. Contains 329968 sequences. (Running on oeis4.)