Many people know of perfect numbers, which in simple terms are defined as natural numbers whose proper divisors add up to the number itself (for example, the factors of 6 are 1,2, and 3, and 1+2+3 = 6). The more technical definition for a perfect number is that when the sigma function is applied to such a number, the result is double the number itself. That is, σ(n) = 2n. A triply perfect number is a number for which σ(n) = 3n.

There are six known triply perfect numbers, and it has been conjectured that there are no others. I humbly present them here, as well as the calculations used to demonstrate their triply-perfectness:

  • 120: σ(120) = σ(2^3*3*5) = 15 * 4 * 6 = 2^3 * 3^2 * 5 = 3*120

  • 672: σ(672) = σ(2^5*3*7) = 63*4*8 = 2^5*3^2*7 = 3*672

  • 523776: σ(523776) = σ(2^9*3*11*31) = 1023*4*12*32=2^9*3^2*11*31 = 3*523776

  • 459818240: σ(459818240) = σ(2^8*5*7*19*37*73) = 511*6*8*20*38*74=2^8*3*5*7*19*37*73 = 3*459818240

  • 1476304896: σ(1476304896) = σ(2^13*3*11*43*127)=(2^14-1)*4*12*44*128 = 2^13*3^2*11*43*127 = 3*147304896

  • 51001180160: σ(51001180160) = σ(2^14*5*7*19*31*151) = (2^15-1)*6*8*20*32*152 = 2^14*3*5*19*31*151 = 3*51001180160

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