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

1,1

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.

Links

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

Example

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.

Mathematica

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

Prog

(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

Crossrefs

Intersection of A005117 and A134605.

Sequence in context: A347040 A252547 A211601 * A199745 A200145 A179271

Adjacent sequences: A379776 A379777 A379778 * A379781 A379782 A379783

Keyword

nonn,new

Author

Charles L. Hohn, Jan 02 2025

Status

approved