42 (number) - Mathematics

Mathematics

• Given 27 same-size cubes whose nominal values progress from 1 to 27, a 3×3×3 magic cube can be constructed such that every row, column, and corridor, and every diagonal passing through the center, is composed of 3 cubes whose sum of values is 42.

• Forty-two is a pronic number and an abundant number; its prime factorization 2 · 3 · 7 makes it the second sphenic number and also the second of the form { 2 · 3 · r }. As with all sphenic numbers of this form, the aliquot sum is abundant by 12. 42 is also the second sphenic number to be bracketed by twin primes; 30 is also a pronic number and also rests between two primes. 42 has a 14 member aliquot sequence 42, 54, 66, 78, 90, 144, 259, 45, 33, 15, 9, 4, 3, 1, 0 and is itself part of the aliquot sequence commencing with the first sphenic number 30. Further, 42 is the 10th member of the 3-aliquot tree.
• It is a Catalan number. Consequently; 42 is the number of noncrossing partitions of a set of five elements, the number of triangulations of a heptagon, the number of rooted ordered binary trees with six leaves, the number of ways in which five pairs of nested parentheses can be arranged, etc.
• It is conjectured to be the scaling factor in the leading order term of the "sixth moment of the Riemann zeta function". In particular, Conrey & Ghosh have conjectured

  where the infinite product is over all prime numbers, p.

• It is the third pentadecagonal number. It is a meandric number and an open meandric number.
• 42 is a Størmer number.
• 42 is a perfect score on the USA Math Olympiad (USAMO) and International Mathematical Olympiad (IMO).
• In base 10, this number is a Harshad number and a self number, while it is a repdigit in base 4 (as 222).
• 42 is the only known value that is the number of sets of four distinct positive integers a,b,c,d, each less than the value itself, such that ab-cd, ac-bd, and ad-bc are each multiples of the value. Whether there are other values remains an open question.
• 42 is a (2,6)-perfect number (super-multiperfect), as