We define factoriangular numbers as sum of corresponding factorials and triangular numbers, that is, n! + n(n+1)/2, n is a natural number. We listed the first few factoriangular numbers: 2, 5, 12, 34, 135, 741, 5068, 40356, 362925, 3628855, 39916866, 479001678, 6227020891, 87178291305, 1307674368120 and notice that each is either a deficient or an abundant number. Is there any such number that is a perfect number? Or, how can we prove that no factoriangular number is a perfect number?
It is easy to show that n!+n(n+1)/2 is never an even perfect number. It is known that every even perfect number has form 2p-1(2p-1), where p and 2p-1 are primes. If n!+n(n+1)/2=2p-1(2p-1), then n(n+1)
It is easy to show that n!+n(n+1)/2 is never an even perfect number. It is known that every even perfect number has form 2p-1(2p-1), where p and 2p-1 are primes. If n!+n(n+1)/2=2p-1(2p-1), then n(n+1)
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