A379780 Composite squarefree integers for which the sum of the squares of their factors is a square.

2145, 2730, 4305, 6545, 9030, 10545, 11935, 13398, 13585, 19695, 20202, 20559, 20735, 21318, 23345, 25530, 25665, 26070, 27030, 27265, 28842, 30849, 34255, 35105, 37345, 38335, 40170, 42159, 45105, 47215, 53382, 56145, 57505, 58938, 59334, 60630, 61761, 63921

Offset: 1

Author

Charles L. Hohn, Jan 02 2025

Keywords

• nonn

Comments

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

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.

Examples

			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.
		

Links

• Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Crossrefs

Intersection of A005117 and A134605.

Programs

Mathematica

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

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)))

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