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A350040
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Number of integer-sided right triangles with hypotenuse A009003(n).
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1
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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
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OFFSET
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1,7
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COMMENTS
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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
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
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LINKS
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PROG
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(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
(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
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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