In Mathematics
33 is the largest positive integer that cannot be expressed as a sum of different triangular numbers. It is also the smallest odd repdigit that is not a prime number.
33 is the eighth distinct semiprime comprising the prime factors (3 • 11). Its aliquot sum is 15; itself a discrete semiprime (3 • 5) in the following Aliquot sequence 33,15,9,4,3,1,0. (Note 33 is the 8th composite number to descend into the prime number 3, the others outside of this sequence being 30,26,16,12) Since 33 is a semiprime with both its prime factors being Gaussian primes, 33 is a Blum integer.
The sum of the first four positive factorials is 33. Adding up the sums of divisors of 1 through 6 yields 33. 33 is the smallest integer such that it and the next two integers all have the same number of divisors.
It is also the first member of the first cluster of three semiprimes (33,34,35); the next such cluster is 85,86,87.
Since the greatest prime factor of 332 + 1 = 1090 is 109, which is obviously more than 33 twice, 33 is a Størmer number.
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