A350040 Number of integer-sided right triangles with hypotenuse A009003(n).

1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 2, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 4, 1, 1, 1, 1, 1, 4, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 2, 4, 1, 1, 2, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 4, 1, 2

Offset: 1

Author

Wesley Ivan Hurt, Dec 11 2021

Keywords

• nonn
• easy

Comments

a(n) mod 3 = 1 for 95.3% of the first 13211 terms, including the 70% where a(n) = 1, and only 4.7% account for the other numbers. Theorem 7 of A. Tripathi (see link below) provides the explanation that 2, 3, 5, 6, etc. are so rare. The term 8 will appear for the first time when the hypotenuse is A006339(8) = 390625. - Ruediger Jehn, Jan 13 2022

All positive integers eventually appear in this sequence. - Charles R Greathouse IV, Jan 13 2022

The normal value of a(n) is roughly log(n)^(log(3)/2). For any fixed k, the asymptotic density of n such that a(n) <= k is 0. The typical a(n) is of the form (x*3^y-1)/2 with x small (because most numbers have only a few primes with exponents > 1). - Charles R Greathouse IV, Jan 13 2022

Links

• Ruediger Jehn, Table of n, a(n) for n = 1..13211
• A. Tripathi, On Pythagorean triples containing a fixed integer, Fib. Q., 46/47 (2008/2009), 331-340.
• Wikipedia, Integer Triangle

Crossrefs

Cf. A006339, A009003, A046080.

Programs

PARI

is_A009003(n)=setsearch(Set(factor(n)[, 1]%4), 1);
f(n) = {my(f = factor(n/(2^valuation(n, 2)))); (prod(k=1, #f~, if ((f[k, 1] % 4) == 1, 2*f[k, 2] + 1, 1)) - 1)/2; } \\ A046080
lista(nn) = apply(f, select(is_A009003, [1..nn])); \\ Michel Marcus, Jan 13 2022

PARI

A046080(n,f=factor(n))=prod(k=if(f[1,1]==2,2,1), #f~, if (f[k,1]%4 == 1, 2*f[k,2] + 1, 1))\2; \\ doesn't handle n = 1, not relevant here
upto(lim)=my(v=List(),u=vectorsmall(lim\=1)); forprimestep(p=5,lim,4, forstep(n=p,lim,p, u[n]=1)); forfactored(n=5,lim, if(u[n[1]], listput(v, A046080(0,n[2])))); u=0; Vec(v) \\ Charles R Greathouse IV, Jan 13 2022

PARI

upto(lim)=my(v=List()); forfactored(n=5,lim\=1, if(vecmin(n[2][,1]%4)==1, listput(v, prod(k=if(n[2][1,1]>2,1,2),#n[2]~, if (n[2][k,1]%4 == 1, 2*n[2][k,2] + 1, 1))\2))); Vec(v) \\ Charles R Greathouse IV, Jan 13 2022