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A334802 Positive integers of the form x^4 - y^4 that have exactly 4 divisors.

%I A334802 #24 Jun 17 2020 02:09:12
%S A334802 15,65,671,3439,12209,102719,113521,178991,246559,515201,1124111,
%T A334802 1342879,2964961,3940399,9951391,21254449,27220159,34209169,45259649,
%U A334802 48986321,70710641,92110289,93084991,125620111,131687681,144402721,201792079,211782751,276694241
%N A334802 Positive integers of the form x^4 - y^4 that have exactly 4 divisors.
%C A334802 If a(n) = pq, where p > q are both prime, then p is the hypotenuse and q is a leg of a primitive Pythagorean triple. (x^4-y^4 = (x^2+y^2)(x+y)(x-y), hence x-y=1 and x^2+y^2 and x+y are both prime. Note that x^2+y^2 can never be (x+y)^2 so a(n) is never the cube of a prime.)
%H A334802 Robert Israel, <a href="/A334802/b334802.txt">Table of n, a(n) for n = 1..10000</a>
%F A334802 a(n) = (b(n)+1)^4 - b(n)^4 with b(n)=A068501(n).
%F A334802 a(n) = A048161(n)*A067756(n).
%e A334802 2^4 - 1^4 = 15 = 3*5 and (3, 4, 5) is a Pythagorean triple, so 15 is a term.
%e A334802 6^4 - 5^4 = 671 = 11*61 and (11, 60, 61) is a Pythagorean triple, so 671 is a term.
%p A334802 f:= proc(y) if isprime(2*y+1) and isprime(2*y^2 + 2*y+1) then (2*y+1)*(2*y^2+2*y+1) fi end proc:
%p A334802 map(f, [$1..1000]); # _Robert Israel_, Jun 16 2020
%t A334802 Select[(#^4 - (#-1)^4) & /@ Range[420], DivisorSigma[0, #] == 4 &] (* _Giovanni Resta_, May 12 2020 *)
%Y A334802 Cf. A068501.
%Y A334802 Intersection of A030513 and A147857.
%K A334802 nonn
%O A334802 1,1
%A A334802 _C. Kenneth Fan_, May 12 2020