A140362 Semiprimes pq that divide the sum of the squares of their divisors, 1+p^2+q^2+(pq)^2.

10, 65, 20737

Offset

1,1

Comments

6 is the smallest integer n which is the product of two distinct primes and which divides the sum of the cubes of the divisors of n. Are there other numbers with this property?

Using Pell equations and a Fibonacci identity, Max Alekseyev and I have shown that all terms are the product of prime Fibonacci numbers whose indices are twin primes. The first three terms are Fib(3)*Fib(5), Fib(5)*Fib(7) and Fib(11)*Fib(13). The other two known terms are Fib(431)*Fib(433) and Fib(569)*Fib(571), huge numbers that are in the b-file. The sequence probably has no additional terms. - T. D. Noe, Jul 27 2008

Let a, b, c and d be consecutive odd-indexed Fibonacci numbers. Then it can be proved that 1 + b^2 + c^2 + (bc)^2 = abcd, which shows that bc divides 1 + b^2 + c^2 + (bc)^2. Hence if b and c are prime, then bc is in this sequence. - T. D. Noe, Jul 27 2008

Links

T. D. Noe, Table of n, a(n) for n=1..5

T. Cai, D. Chen, Y. Zhang, Perfect numbers and Fibonacci primes, arXiv:1310.0898 [math.NT], 2013-2014.

T. Cai, D. Chen, Y. Zhang, Perfect numbers and Fibonacci primes (II), arXiv:1406.5684 [math.NT], 2014 (see case m=1 in Table 1).

Example

10 divides (1^2 + 2^2 + 5^2).
65 divides (1^2 + 5^2 + 13^2).
20737 divides (1^2 + 89^2 + 233^2).

Prog

(PARI) isok(n) = sigma(n, 2) - n^2 == 3*n; \\ Michel Marcus, Jun 24 2014

Crossrefs

Cf. A000045, A001605, A046762.

Sequence in context: A307144 A344047 A344060 * A159838 A269679 A211056

Adjacent sequences: A140359 A140360 A140361 * A140363 A140364 A140365

Keyword

nonn,bref

Author

Mohamed Bouhamida, Jul 22 2008, Jul 27 2008

Status

approved