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Applying the Kaprekar routine to 4-digit number reaches 0 for exactly 77 4-digit numbers, while the remainder give 6174 in at most 8 iterations. The value 6174 is sometimes ...

The number k in the expression s(n)=kn for a multiperfect number is called its class.

A number n for which the product of divisors is equal to n^2. The first few are 1, 6, 8, 10, 14, 15, 21, 22, ... (OEIS A007422).

A positive integer n is kth powerfree if there is no number d such that d^k|n (d^k divides n), i.e., there are no kth powers or higher in the prime factorization of n. A ...

A proper factor of a positive integer n is a factor of n other than 1 or n (Derbyshire 2004, p. 32). For example, 2 and 3 are positive proper factors of 6, but 1 and 6 are ...

A 10-digit number satisfying the following property. Number the digits 0 to 9, and let digit n be the number of ns in the number. There is exactly one such number: 6210001000.

sum_(n=1)^(infty)1/(phi(n)sigma_1(n)) = product_(p prime)(1+sum_(k=1)^(infty)1/(p^(2k)-p^(k-1))) (1) = 1.786576459... (2) (OEIS A093827), where phi(n) is the totient function ...

Consecutive Smith numbers. The first few Smith brothers are (728, 729), (2964, 2965), (3864, 3865), (4959, 4960), ... (OEIS A050219 and A050220).

A 3-multiperfect number P_3. Six sous-doubles are known (120, 672, 523776, 459818240, 1476304896, and 51001180160; OEIS A005820), and these are believed to comprise all ...

A 4-multiperfect number P_4. 36 sous-triples are known (30240, 32760, 2178540, 23569920, ...; OEIS A027687), and these are believed to comprise all sous-triples.