Sequences

Convolution of natural numbers n >= 1 with Lucas numbers L(k)(A000032) for k >= 2.

The Loh-Shu 3 X 3 magic square, lexicographically largest variant when read by columns.

Convolution of natural numbers n >= 1 with Lucas numbers L(k) (A000032) for k >= 3.

Convolution of positive integers n with Lucas numbers L(k)(A000032) for k >= 4.

Number of standard permutations of [ a_1..a_n b_1..b_n ] (b_i is not immediately followed by a_i, for all i).

a(n) = 2*n^2 + 3*n + 3.

Convolution of natural numbers n >= 1 with Lucas numbers L(k) for k >= -4.

Convolution of natural numbers n >= 1 with Lucas numbers L(k) for k >= -2.

Trimorphic numbers: n^3 ends with n. Also m-morphic numbers for all m > 5 such that m-1 is not divisible by 10 and m == 3 (mod 4).

Triangle read by rows: T(k,j) = ((2*j+1)/(k+1))*binomial(2*j,j)*binomial(2*k-2*j,k-j).

Orders of certain groups associated with K3 surfaces and M24.

Numbers of fixed points of elements of group M24.

Numbers of fixed points of elements of group M24.

Finite sequence associated with M24.

Finite sequence associated with M24.

Critical dimensions for N-modular lattices.

Critical dimensions for N-modular lattices.

Numbers that can be expressed as the product of two 2-digit numbers in at least one way.

Numbers that can be expressed as the product of two 3-digit numbers in at least one way.

Numbers that can be expressed as the product of three 2-digit numbers.

Number of numbers d dividing n such that d >= 3 and n/d <= d-2.

Sum of odd divisors of n < sqrt(n) = sum of even divisors of n < sqrt(n).

Highly factorable numbers: numbers with a record number of proper factorizations.

Number of proper factorizations of the numbers with a record number of proper factorizations.

Smallest number > 1 equal to sum of n-th powers of its base-3 digits, or 0 if no such number exists (written in base 10).

Smallest number > 1 equal to sum of n-th powers of its base-4 digits, or 0 if no such number exists (written in base 10).

Smallest number > 1 equal to sum of n-th powers of its base-5 digits, or 0 if no such number exists (written in base 10).

Smallest number > 1 equal to sum of n-th powers of its base-6 digits, or 0 if no such number exists (written in base 10).

Smallest number > 1 equal to sum of n-th powers of its base-7 digits, or 0 if no such number exists (written in base 10).

Smallest number > 1 equal to sum of n-th powers of its base-8 digits, or 0 if no such number exists (written in base 10).

Smallest number > 1 equal to sum of n-th powers of its base-9 digits, or 0 if no such number exists (written in base 10).

Triangle of coefficients of certain polynomials (exponents in decreasing order).

Number of twin primes < 2^n.

a(n) = prime(2^n).

Numbers k of the form 2^i*3^j, where i and j >= 1.

Numbers whose prime factors are 2 and 5.

Numbers whose prime factors are 2 and 7.

Numbers whose prime factors are 2 and 11.

Numbers whose prime factors are 3 and 5.

Numbers whose prime factors are 3 and 7.

Numbers whose prime factors are 5 and 7.

Integers k such that j(k)*phi(k) = sigma(k), where j(n) = A033831(n).

Integers k such that j(k)*d(k)=phi(k), where j = A033831.

Numbers k such that j(k)*ud(k)=phi(k), j(n) = A033831, ud(n) = number of unitary divisors of n (A034444).

Numbers k such that j(k)*phi(k) = s(phi(k)), where j(k) = A033831(k), s(k) = sigma(k) - k.

Numbers k such that j(k)*phi(k) = sigma(phi(k)), j(k) = A033831(k).

Numbers n such that j(n)*phi(n) = usigma(n), where j(n) = A033831, usigma(n) = sum of unitary divisors of n (A034448).

Numbers k such that j(k) + ud(k) = d(k), where j(k) = A033831, ud(k) = number of unitary divisors of k (A034444), d(k) = number of divisors of k (A000005).

Numbers k such that A033831(k) = A034444(k) where A034444(k) = number of unitary divisors of k.

Sort-then-add sequence: a(n+1) = a(n) + sort(a(n)).