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%I #29 Jul 12 2023 11:07:43
%S 5,34,53,90,120,440,780,1954,120994,140453,28813276834
%N Numbers n such that n-tau(n), phi(n) and n form a Heronian triangle, where tau=A000005 is the number of divisors and phi=A000010 the totient.
%C For all n, n > tau(n) and n > phi(n) and if n is prime then n-tau(n) = n-2 and phi(n) = n-1. So n = 5 gives the triangle {3, 4, 5} which is a primitive Pythagorean triangle and this is the only one. Other Pythagorean triangles are {30, 16, 34} and {756, 192, 780}, the remainder are only Heronian.
%C It is not known if this sequence is infinite. Prime numbers in the sequence are 5, 53 and 140453 and generate triangles {3, 4, 5}, {51, 52, 53} and {140451, 140452, 140453}.
%C If n = 2p where p is prime then n-tau(n) = n-4 and phi(n) = n/2-1. So n = 34 gives the triangle {16, 30, 34}. Similar numbers in this sequence are a(8), a(9) and a(11). See A272365 for generating Heronian triangles with sides n, n-4, n/2-1.
%C a(12) > 2*10^12. - _Giovanni Resta_, Apr 14 2016
%C Next prime value of a(n) after 140453 is > 2*10^5719. See A003500 for generating Heronian triangles with consecutive sides. - _Frank M Jackson_, Apr 19 2016
%C A003500(n)+1 is a member of this sequence iff it is prime. Also A272365(n) is a member of this sequence iff A272365(n)/2 is prime. - _Frank M Jackson_, Apr 29 2016
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/DivisorFunction.html">Divisor Function</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/TotientFunction.html">Totient Function</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/HeronianTriangle.html">Heronian Triangle.</a>
%e a(2) = 34 because the triangle so formed has sides 30, 16, 34. It is Heronian with integer area 240 and is also Pythagorean. It is the second Heronian triangle.
%e The triangle corresponding to a(11) has sides n = 28813276834, n-tau(n) = 28813276830, phi(n) = 14406638416, and area 200960614753814018640.
%t triples[n_] := ({a, b, c}={n-DivisorSigma[0, n], EulerPhi[n], n}; s=(a+b+c)/2; If[a+b>c&&IntegerQ[Sqrt[s(s-a)(s-b)(s-c)]], {a, b, c}, {}]); lst={}; Do[If[triples[n]!={}, AppendTo[lst, Last[triples[n]]]], {n, 1, 200000}]; lst
%Y Cf. A000005, A000010, A003500, A046022, A049820, A084820, A272365.
%K nonn,more
%O 1,1
%A _Frank M Jackson_, Jan 29 2016
%E a(11) from _Giovanni Resta_, Apr 14 2016