78 (number)

← 77 78 79 →
← 70 71 72 73 74 75 76 77 78 79 → List of numbers Integers ← 0 10 20 30 40 50 60 70 80 90 →
Cardinal	seventy-eight
Ordinal	78th (seventy-eighth)
Factorization	2 × 3 × 13
Divisors	1, 2, 3, 6, 13, 26, 39, 78
Greek numeral	ΟΗ´
Roman numeral	LXXVIII, lxxviii
Binary	10011102
Ternary	22203
Senary	2106
Octal	1168
Duodecimal	6612
Hexadecimal	4E16

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

In mathematics

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). ^[1]
• 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. ^[2]
• the 12th triangular number. ^[3]
• a palindromic number in bases 5 (303₅), 7 (141₇) ^[4] , 12 (66₁₂) ^[5] , 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. ^[6]
• 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: ${\displaystyle 8^{2}+3^{2}+2^{2}+1^{2}}$, ${\displaystyle 7^{2}+4^{2}+3^{2}+2^{2}}$ or ${\displaystyle 6^{2}+5^{2}+4^{2}+1^{2}}$ (see image). ^{[7] [8]}

77 and 78 form a Ruth–Aaron pair.

References

1. "Sloane's A007304 : Sphenic numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-05-29.
2. Sloane, N. J. A. (ed.). "SequenceA005835(Pseudoperfect (or semiperfect) numbers n: some subset of the proper divisors of n sums to n.)". The On-Line Encyclopedia of Integer Sequences . OEIS Foundation.
3. "A000217 - OEIS". oeis.org. Retrieved 2024-11-28.
4. Sloane, N. J. A. (ed.). "SequenceA249156". The On-Line Encyclopedia of Integer Sequences . OEIS Foundation.
5. Sloane, N. J. A. (ed.). "SequenceA029957". The On-Line Encyclopedia of Integer Sequences . OEIS Foundation.
6. "Sloane's A059756 : Erdős-Woods numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-05-29.
7. Sloane, N. J. A. (ed.). "SequenceA025386(Numbers that are the sum of 4 distinct nonzero squares in 2 or more ways.)". The On-Line Encyclopedia of Integer Sequences . OEIS Foundation.
8. Sloane, N. J. A. (ed.). "SequenceA025378(Numbers that are the sum of 4 distinct nonzero squares in exactly 3 ways.)". The On-Line Encyclopedia of Integer Sequences . OEIS Foundation.