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The first: 1, 8, 27, 64, 125, 216, 343, 512, 729, 1000, 1331, 1728 (sequence A000578 in the OEIS). A perfect power has a common divisor m > 1 for all multiplicities (it is of the form a m for some a > 1 and m > 1).
An overlapping circles grid is a geometric pattern of repeating, ... (8-1) 19-circle (27-8) 37-circle (64-27) 61-circle (125-64) 91-circle (216-125)
of numbers that, when written as a ternary number, use only the digits 0 and 1. This sequence is the lexicographically first infinite Salem–Spencer set. [5] Another infinite Salem–Spencer set is given by the cubes. 0, 1, 8, 27, 64, 125, 216, 343, 512, 729, 1000, ... (sequence A000578 in the OEIS)
Name First elements Short description OEIS Mersenne prime exponents : 2, 3, 5, 7, 13, 17, 19, 31, 61, 89, ... Primes p such that 2 p − 1 is prime.: A000043 ...
With the exceptions of 1, 8 and 144 (F 1 = F 2, F 6 and F 12) every Fibonacci number has a prime factor that is not a factor of any smaller Fibonacci number (Carmichael's theorem). [57] As a result, 8 and 144 (F 6 and F 12) are the only Fibonacci numbers that are the product of other Fibonacci numbers. [58]
For example, the third triangular number is (3 × 2 =) 6, the seventh is (7 × 4 =) 28, the 31st is (31 × 16 =) 496, and the 127th is (127 × 64 =) 8128. The final digit of a triangular number is 0, 1, 3, 5, 6, or 8, and thus such numbers never end in 2, 4, 7, or 9. A final 3 must be preceded by a 0 or 5; a final 8 must be preceded by a 2 or 7.
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Second edition of the book. Neil Sloane started collecting integer sequences as a graduate student in 1964 to support his work in combinatorics. [8] [9] The database was at first stored on punched cards.