Sequences

Triangle of D-analogs of Stirling numbers of the 2nd kind.

Triangle of D-analogs of Stirling numbers of first kind.

Triangle of D-analogs of Stirling numbers of first kind, rows reversed.

D-analogs of Bell numbers.

Number of edges in the Hasse diagrams for the D-analogs of the partition lattices.

Numbers k such that gcd(phi(k), k-1) = number of divisors of k.

Numbers k such that gcd(phi(k), k-1) = number of distinct prime factors of (k-1).

Numbers k such that gcd(phi(k), k-1) = number of divisors of (k-1).

Composite integers k such that gcd(phi(k), k - 1) > 1.

Numbers k such that phi(k) is a perfect square.

Numbers k such that phi(k) is a perfect cube.

Even numbers k such that phi(k) and k-1 are distinct and have a common factor > 1.

phi(n) is equal to the sum of prime factors and exponents of n-1.

Numbers k such that phi(k) is equal to the product of (the sum of prime factors and the sum of exponents) of k-1.

Number of new fractions m/n, where m is prime, (m,n)=1 and "new" means the value of mn has not occurred before.

Number of new fractions m/n, where m is 1 or composite, (m,n) = 1 and "new" means the value of mn has not occurred before.

Integers m such that phi(m) is equal to the sum of (the product of prime factors) and (the product of exponents) of m-1.

phi(n) is equal to the sum of prime factors and exponents of n+1.

Numbers k such that phi(k) is equal to A008473(k).

Numbers k such that phi(k) is equal to A008473(k-1).

Numbers k such that phi(k) is equal to A008473(k+1).

phi(n) is equal to the sum of the prime-power components of n-1.

phi(n) is equal to the product of the dual prime-power components of n-1 (i.e., A008477(n-1)).

phi(n) is equal to the product of the dual prime-power components of n.

Numbers n such that phi(n) is equal to the multiplicative projection of n.

phi(a(n)) is equal to the multiplicative projection of (a(n)-1).

Primes p such that p-1 is squarefree.

Numbers k such that phi(k) is equal to the product of (the sum of prime factors and the sum of exponents) of k.

Integers k such that phi(k) is equal to the product of (the sum of prime factors and the sum of exponents) of k+1.

Prime numbers prefixed with a '1'.

Sequence arising in search for Legendre sequences.

Sequence arising in search for Legendre sequences.

Sequence arising in search for Legendre sequences.

Sequence arising in search for Legendre sequences.

Sequence arising in search for Legendre sequences.

Sequence arising in search for Legendre sequences.

Triangle of numbers of Dyck paths.

Triangle read by rows: numbers of Dyck paths.

For S a subset of [ n ] = {1,2,3,...n}, let B_S = {x+y : x,y in S, x<y}; then a(n) is maximal cardinality of B_S intersect B_{[ n ]-S}.

Column 1 of Inverse partition triangle A038498.

Column 2 of Inverse partition triangle A038498.

Column 3 of Inverse partition triangle A038498.

Column 4 of inverse partition triangle A038498.

Column 5 of Inverse partition triangle A038498.

Matrix cube of partition triangle A008284.

Matrix 4th power of partition triangle A008284.

Matrix 5th power of partition triangle A008284.

Shifts left under transform T where Ta is product of Partition Triangle A008284 with a.

For n > 1, a(n) doubles under the transform T, where Ta is the matrix product of partition triangle A008284 with a, with a(1) = 1.

Matrix square of Stirling2 triangle A008277: 2-levels set partitions of [n] into k first-level subsets.