Triply Perfect Numbers

created by srkorn

(idea) by srkorn (2.4 y) (print) ? (I like it!) Thu Mar 01 2001 at 22:22:57

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

printable version
chaos

sigma function	There is no you	perfect number	26
Lucas number	conjecture	Convert any number to decimal	Node for the Ages
amicable numbers

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