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  2. Table of prime factors - Wikipedia

    en.wikipedia.org/wiki/Table_of_prime_factors

    The first in decimal: 4, 6, 8, 9, 12, 18, 20, 22, 24, 26, 28, 30 (sequence A046760 in the OEIS). An economical number has been defined as a frugal number, but also as a number that is either frugal or equidigital.

  3. Prime number theorem - Wikipedia

    en.wikipedia.org/wiki/Prime_number_theorem

    Graph of the number of primes ending in 1, 3, 7, and 9 up to n for n < 10 000. Another example is the distribution of the last digit of prime numbers. Except for 2 and 5, all prime numbers end in 1, 3, 7, or 9. Dirichlet's theorem states that asymptotically, 25% of all primes end in each of these four digits.

  4. Integer partition - Wikipedia

    en.wikipedia.org/wiki/Integer_partition

    For example, 4 can be partitioned in five distinct ways: 4 3 + 1 2 + 2 2 + 1 + 1 1 + 1 + 1 + 1. The only partition of zero is the empty sum, having no parts. The order-dependent composition 1 + 3 is the same partition as 3 + 1, and the two distinct compositions 1 + 2 + 1 and 1 + 1 + 2 represent the same partition as 2 + 1 + 1.

  5. Fundamental theorem of arithmetic - Wikipedia

    en.wikipedia.org/wiki/Fundamental_theorem_of...

    This representation is called the canonical representation [10] of n, or the standard form [11] [12] of n. For example, 999 = 3 3 ×37, 1000 = 2 3 ×5 3, 1001 = 7×11×13. Factors p 0 = 1 may be inserted without changing the value of n (for example, 1000 = 2 3 ×3 0 ×5 3).

  6. Least common multiple - Wikipedia

    en.wikipedia.org/wiki/Least_common_multiple

    A multiple of a number is the product of that number and an integer. For example, 10 is a multiple of 5 because 5 × 2 = 10, so 10 is divisible by 5 and 2. Because 10 is the smallest positive integer that is divisible by both 5 and 2, it is the least common multiple of 5 and 2.

  7. Persistence of a number - Wikipedia

    en.wikipedia.org/wiki/Persistence_of_a_number

    The additive persistence of 2718 is 2: first we find that 2 + 7 + 1 + 8 = 18, and then that 1 + 8 = 9. The multiplicative persistence of 39 is 3, because it takes three steps to reduce 39 to a single digit: 39 → 27 → 14 → 4. Also, 39 is the smallest number of multiplicative persistence 3.

  8. Explicit formulae for L-functions - Wikipedia

    en.wikipedia.org/wiki/Explicit_formulae_for_L...

    The other terms also correspond to zeros: The dominant term li(x) comes from the pole at s = 1, considered as a zero of multiplicity −1, and the remaining small terms come from the trivial zeros. This formula says that the zeros of the Riemann zeta function control the oscillations of primes around their "expected" positions.

  9. Multiplicity (mathematics) - Wikipedia

    en.wikipedia.org/wiki/Multiplicity_(mathematics)

    Graph of x 3 + 2x 2 − 7x + 4 with a simple root (multiplicity 1) at x=−4 and a root of multiplicity 2 at x=1. The graph crosses the x axis at the simple root. It is tangent to the x axis at the multiple root and does not cross it, since the multiplicity is even.