cp's OEIS Frontend

a((A003961(n) + 1) / 2) = n and A003961(a(n)) = 2*n - 1 for all n. If the sequence is indexed by odd numbers only, it becomes multiplicative. In this variant sequence, denoted b, even indices don't exist, and we get b(1) = a(1) = 1, b(3) = a(2) = 2, b(5) = 3, b(7) = 5, b(9) = 4 = b(3) * b(3), ... , b(15) = 6 = b(3) * b(5), and so on. This property can also be stated as: a(x) * a(y) = a(((2x - 1) * (2y - 1) + 1) / 2) for x, y > 0. - Reinhard Zumkeller [re-expressed by Peter Munn, May 23 2020]

Not multiplicative in usual sense - but letting m=2n-1=product_j (p_j)^(e_j) then a(n)=a((m+1)/2)=product_j (p_(j-1))^(e_j). - Henry Bottomley, Apr 15 2005

From Antti Karttunen, Jul 25 2016: (Start)

Several permutations that use prime shift operation A064989 in their definition yield a permutation obtained from their odd bisection when composed with this permutation from the right. For example, we have:

A243505(n) = A122111(a(n)).

A243065(n) = A241909(a(n)).

A244153(n) = A156552(a(n)).

A245611(n) = A243071(a(n)).

(End)

If n >= 2, n occurs in column a(n) of A246278.

By convention, a(1) = 0 because 1 does not occur in A246278.

This is permutation of natural numbers larger than 1.

From Antti Karttunen, Dec 19 2014: (Start)

If we assume here that a(1) = 1 (but which is not explicitly included because outside of the array), then A252460 gives an inverse permutation. See also A249741.

For navigating in this array:

A055396(n) gives the row number of row where n occurs, and A078898(n) gives its column number, both starting their indexing from 1.

A250469(n) gives the number immediately below n, and when n is an odd number >= 3, A250470(n) gives the number immediately above n. If n is a composite, A249744(n) gives the number immediately left of n.

First cube of each row, which is {the initial prime of the row}^3 and also the first number neither a prime or semiprime, occurs on row n at position A250474(n).

(End)

The n-th row contains the numbers whose least prime factor is the n-th prime: A020639(T(n,k)) = A000040(n). - Franklin T. Adams-Watters, Aug 07 2015

Permutation A249817 preserves the smallest prime factor of n, i.e., A055396(A249817(n)) = A055396(n), in other words, keeps all the terms that appear on any row of A246278 on the same row of A083221. Permutations in this table are induced by changes that A249817 does onto each row of the latter table, thus permutation on row r of this table can be used to sort row r of A246278 into ascending order. I.e., A246278(r, A(r,c)) = A083221(r,c) [the corresponding row in the Sieve of Eratosthenes, where each row appears in monotone order].

The multi-set of cycle-sizes of permutation A249817 is a disjoint union of cycle-sizes of all permutations in this array. For example, A249817 has a 7-cycle (33 39 63 57 99 81 45) which originates from the 7-cycle (6 7 11 10 17 14 8) of A064216, which occurs as the second row in this table.

On each row, 4 is the first composite number (and the first term less than previous, apart from row 1), and on row n it occurs in position A250474(n). This follows because A001222(A246277(n)) = A001222(n)-1 and because on each row of A083221 (see A083140) all terms between the square of prime (second term on each row) and the first cube (of the same prime, this cube mapping in this array to 4) are nonsquare semiprimes (A006881), this implies that the corresponding terms in this array must be primes.

Also, as the smaller prime factor of the terms on row n of A083221 is constant, A020639(n), and for all i < j: A246277(p_{i} * p_{j}) < A246277(p_i * p_{j+1}), the primes on any row appear in monotone order.

Position of the first composite number (which is always 4) on row n of A249821. The fourth column of A249822.

Position of the first nonfixed term on row n of arrays of permutations A251721 and A251722.

According to the definition, this is the number of multiples of prime(n) below prime(n)^3 (and thus, the number of numbers below prime(n)^2) which do not have a smaller factor than prime(n). That is, the numbers remaining below prime(n)^2 after deleting all multiples of primes less than prime(n), as is done by applying the first n-1 steps of the sieve of Eratosthenes (when the first step is elimination of multiples of 2). This explains that the first differences are a(n+1)-a(n) = A050216(n)-1 for n>1, and a(n) = A054272(n)+2. - M. F. Hasler, Dec 31 2014

These are the "first differences" between permutations of array A249822, in a sense that by composing the first k rows of this array [from right to left, as in a(n) = row_k(...(row_2(row_1(n))))], one obtains row k+1 of A249822.

On row n the first non-fixed term is A250474(n+1) at position A250474(n), i.e., on row 1 it is 5 at n=4, on row 2 it is 9 at n=5, on row 3 it is 14 at n=9, etc. All the previous A250473(n) terms are fixed.

a(n) tells how many columns off A003961(n) is from the column where n is in square array A083221 (Cf. A083140, the sieve of Eratosthenes. The column index of n in that table is given by A078898(n)).

a(n) = Length of the initial fixed portion of the permutation at row n of A251721 & A251722.

Also, for n > 1, the length of the maximal increasing prefix on row n of A249821.