Digital Library of Mathematical Functions:

§20.12(i) Number Theory ‣ §20.12 Mathematical Applications ‣ Applications ‣ Chapter 20 Theta Functions
Chapter 27 Functions of Number Theory

Online Encyclopedia of Integer Sequences:

Euler totient function phi(n): count numbers <= n and prime to n.
From von Staudt-Clausen representation of Bernoulli numbers: a(n) = Bernoulli(2n) + Sum_{(p-1)|2n} 1/p.
Numbers that are the sum of 2 nonzero squares.
pi(n), the number of primes <= n. Sometimes called PrimePi(n) to distinguish it from the number 3.14159...
Wieferich primes: primes p such that p^2 divides 2^(p-1) - 1.
Least positive primitive root of n-th prime.
Primes of the form 4*k + 3.
Primes congruent to 1 or 2 modulo 4; or, primes of form x^2 + y^2; or, -1 is a square mod p.
Goldbach conjecture: number of decompositions of 2n into ordered sums of two odd primes.
Primes of the form k^2 + 1.
Markoff (or Markov) numbers: union of positive integers x, y, z satisfying x^2 + y^2 + z^2 = 3*x*y*z.
Heegner numbers: imaginary quadratic fields with unique factorization (or class number 1).
Numbers k such that k^2 + 1 is prime.
Decimal expansion of the twin prime constant C_2 = Product_{ p prime >= 3 } (1-1/(p-1)^2).
Denominators of Bernoulli numbers B_0, B_1, B_2, B_4, B_6, ...
Wilson primes: primes p such that (p-1)! == -1 (mod p^2).
Möbius (or Moebius) function mu(n). mu(1) = 1; mu(n) = (-1)^k if n is the product of k different primes; otherwise mu(n) = 0.
Numbers n such that sqrt(-1) mod n exists; or, numbers that are primitively represented by x^2 + y^2.
From Euler’s Pentagonal Theorem: coefficient of q^n in Product_{m>=1} (1 - q^m).
Number of (positive) squarefree numbers < n.
Numbers that are not the sum of 2 primes.
Number of integer solutions to x^2 + y^2 <= n excluding (0,0).
Exponential of Mangoldt function M(n): a(n) = 1 unless n is a prime or prime power, in which case a(n) = that prime.
a(n) = 10^n - 9^n.
Waring’s problem: least positive integer requiring maximum number of terms when expressed as a sum of positive n-th powers.
Denominator of Sum_{p prime, p-1 divides 2*n} 1/p.
a(n) = tau(n)^2, where tau(n) = A000005(n).
Floor of the Chebyshev function theta(n): a(n) = floor(Sum_{primes p <= n } log(p)).
Nearest integer to Sum_{k=1..n} log(k) = log(n!).
Number of divisors d of 2n satisfying (d+1) = prime or number of prime factors of the denominator of the even Bernoulli numbers.
Sum of divisors of n that are not divisible by 4.
Nearest integer to n*log(n).
Numbers m such that the Bernoulli number B_{2*m} has denominator 30.
Numbers m such that the Bernoulli number B_m has denominator 30.
Numbers m such that the Bernoulli number B_{2*m} has denominator 42.
As p runs through the primes == 1 mod 3, sequence gives Bernoulli(2p) - 1/6.
Number of ways of writing 2n+1 as p + q + r where p, q, r are primes with p <= q <= r.
a(n) = Product_{i=2..n} (prime(i) - 2).
a(n) = Product_{i=3..n} (prime(i) - 3).
a(n) = Product_{i=4..n} (prime(i)-5), where prime(i) is i-th prime.
Decimal expansion of alpha(2) = Sum_{i>0} prime(i)*2^(-i^2).
Denominator of 1*2*4*6*...*(prime(n-1)-1) / (2*3*5*7*...*prime(n-1)).
As p runs through the primes >= 5, sequence gives { numerator of Sum_{k=1..p-1} 1/k } / p^2.
Simple quadratic fields (i.e., with a unique prime factorization).
Numbers n such that { x +- 2^k : 0 < k < 4 } are primes, where x = 210*n - 105.
Find smallest k such that k^n is a sum of n n-th powers, say k^n = T(n,1)^n + ... + T(n,n)^n. Sequence gives triangle of successive rows T(n,1), ..., T(n,n). T(n,1) = ... = T(n,n) = 0 indicates no solution exists.
Numerators in partial products of the twin prime constant.
Denominators in partial products of the twin prime constant.
Triangle of falling factorials, read by rows: T(n, k) = n*(n-1)*...*(n-k+1), n > 0, 1 <= k <= n.
Waring’s problem: conjectured values for G(n), the smallest number m such that every sufficiently large number is the sum of at most m n-th powers of positive integers.
Triangle in which n-th row lists all partitions of n, in graded reverse lexicographic ordering.
Number of primes of the form x^2 + 1 < 10^n.
a(n)^2 + 1 is largest prime of the form x^2 + 1 <= 10^n.
a(n) is the largest prime of the form x^2 + 1 <= 10^n.
a(n) = number of primes of the form x^2 + 1 <= 2^n.
a(n)^2 + 1 is largest prime of the form x^2 + 1 <= 2^n.
a(n) is the largest prime of the form x^2 + 1 <= 2^n.
Smallest number expressible as the sum of three 4th powers in at least n ways.
a(n) = floor( prime(n)/log(n) ).
Numbers n such that abs(d(n) - log(n) + 1 - 2*gamma) is a decreasing sequence, where d(n) is the number of divisors A000005(n) and gamma is Euler’s constant A001620.
Moebius mu sequence for real quadratic extension sqrt(2).
a(n) = 0 for even n, a(n) = (-1)^((n-1)/2) for odd n. Periodic sequence 1,0,-1,0,...
Decimal expansion of -x, where x is the real root of f(x) = 1 + (twin_prime(n))x^n.
a(n) = (2^(semiprime(n)-1)) modulo (semiprime(n)^2).
(Number of squarefree numbers <= n) minus round(n/zeta(2)).
n such that the Moebius function take successively, from n, the values -1,0,-1,0,-1,0.
Smallest m such that the Moebius function takes successively, from m, n values 1,0,1,0,... ending with 1 or 0.
Numerator of product_{k=1..n-1} (1 + 1/prime(k)).
Denominator of product_{k=1..n-1} (1 + 1/prime(k)).
Values of D for which the imaginary quadratic field Q[ sqrt(-D) ] is norm-Euclidean.
Integers k such that 8*k + 1 is a prime or a square.