78 (number)

Template:Refimprove Template:Infobox number 78 (seventy-eight) is the natural number following 77 and preceding 79.

In mathematics

File:Square-sum-78.png

78 as the sum of four distinct nonzero squares

78 is:

• the 5th discrete tri-prime; or also termed Sphenic number, and the 4th of the form (2.3.r).
• an abundant number with an aliquot sum of 90; within an aliquot sequence of nine composite numbers (78, 90,144,259,45,33,15,9,4,3,1,0) to the Prime in the 3-aliquot tree.
• a semiperfect number, as a multiple of a perfect number.
• the 12th triangular number.
• a palindromic number in bases 5 (303₅), 7 (141₇), 12 (66₁₂), 25 (33₂₅), and 38 (22₃₈).
• a Harshad number in bases 3, 4, 5, 6, 7, 13 and 14.
• an Erdős–Woods number, since it is possible to find sequences of 78 consecutive integers such that each inner member shares a factor with either the first or the last member.
• the dimension of the exceptional Lie group E₆ and several related objects.
• the smallest number that can be expressed as the sum of four distinct nonzero squares in more than one way: $8^2+3^2+2^2+1^2$, $7^2+4^2+3^2+2^2$ or $6^2+5^2+4^2+1^2$ (see image).

77 and 78 form a Ruth–Aaron pair.