A379780 - OEIS

%I #10 Jan 03 2025 20:54:58

%S 2145,2730,4305,6545,9030,10545,11935,13398,13585,19695,20202,20559,

%T 20735,21318,23345,25530,25665,26070,27030,27265,28842,30849,34255,

%U 35105,37345,38335,40170,42159,45105,47215,53382,56145,57505,58938,59334,60630,61761,63921

%N Composite squarefree integers for which the sum of the squares of their factors is a square.

%C Also, products of base lengths of Pythagorean hyperrectangles whose base lengths are distinct primes.

%C Observed from a sampling of values up to 10^15 that density approximately halves for each tenfold increase in a(n), though gap sizes between successive terms have high variability.

%H Charles R Greathouse IV, <a href="/A379780/b379780.txt">Table of n, a(n) for n = 1..10000</a>

%e 2145 is included (as a(1), being the smallest such integer) because 2145 = 3 * 5 * 11 * 13 and 3^2 + 5^2 + 11^2 + 13^2 = 18^2.

%t Select[Range[64000],CompositeQ[#]&&SquareFreeQ[#]&&IntegerQ[Sqrt[Total[First/@FactorInteger[#]^2]]]&] (* _James C. McMahon_, Jan 03 2025 *)

%o (PARI) for(t=2, 1000000, if(!issquarefree(t) || isprime(t), next); v=Vec(factor(t)); if(issquare(sum(i=1, #v[1], v[1][i]^2)), print(t)))

%o (PARI) list(lim)=my(v=List()); forsquarefree(n=2145,lim\1, if(issquare(norml2(n[2][,1])) && #n[2][,1]~>1, listput(v,n[1]))); Vec(v) \\ _Charles R Greathouse IV_, Jan 02 2025

%Y Intersection of A005117 and A134605.

%K nonn

%O 1,1

%A _Charles L. Hohn_, Jan 02 2025