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a(n) tells which number in square array A083221 (the sieve of Eratosthenes) is at the same position where n is in array A246278. As both arrays have even numbers as their topmost row and primes as their leftmost column, both sequences are among the fixed points of this permutation.

Equally: a(n) tells which number in array A083140 is at the same position where n is in the array A246279, as they are the transposes of above two arrays.

Odd bisection, A250472, is a permutation of natural numbers. A250479 gives the even bisection.

For odd numbers n >= 3, a(n) = A078898(n)-th number k for which A055396(k) = A055396(n)-1. In other words, a(n) tells which number is located immediately above n in the sieve of Eratosthenes (see A083140, A083221) in the same column of the sieve that contains n.

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.

From Antti Karttunen, Dec 17 2014, further edited Jan 01 & 04 2014: (Start)

Semiprimes p*q, p < q, such that the smallest r for which r^k <= p and q < r^(k+1) [for some k >= 0] is q+1, and thus k = 0. In other words, semiprimes whose both prime factors do not fit (simultaneously) between any two consecutive powers of any natural number r less than or equal to the larger prime factor. This condition forces the larger prime factor q to be greater than the square of the smaller prime factor because otherwise the opposite condition given in A251728 would hold.

Assuming that A054272(n), the number of primes in interval [prime(n), prime(n)^2], is nondecreasing (implied for example if Legendre's or Brocard's conjecture is true), these are also "unsettled" semiprimes that occur in a square array A083221 constructed from the sieve of Eratosthenes, "above the line A251719", meaning that if and only if row < A251719(col) then a semiprime occurring at A083221(row, col) is in this sequence, and conversely, all the semiprimes that occur at any position A083221(row, col) where row >= A251719(col) are in the complementary sequence A251728.

(End)

Semiprimes p*q, p < q, such that b = q+1 is the minimal base with the property that p and q have equal length representations in base b. This was the original definition, which is based primarily on A138510: A138510(A174956(a(n))) = A084127(A174956(a(n))) + 1.

Numbers 7*k such that gcd(k,30) = 1.

Numbers congruent to 7, 49, 77, 91, 119, 133, 161, 203 modulo 210. - Jianing Song, Nov 18 2022

a(n) tells which number in Ludic array A255127 is at the same position where n is in array A083221, the sieve of Eratosthenes. As both arrays have A005843 (even numbers) and A016945 as their two topmost rows, both sequences are among the fixed points of this permutation.

Equally: a(n) tells which number in array A255129 is at the same position where n is in the array A083140, as they are the transposes of above two arrays.

The unit, 1, has the empty prime signature { } (thus not in triangle).

Downwards diagonals:

* Rightmost diagonal: smallest numbers of a given prime signature in increasing order (A025487). This defines the order of signatures used.

This special ordering of prime signatures (by increasing smallest numbers of a given prime signature, A181087) is unrelated to any of the 8 variants of graded lexicographic or colexicographic orderings (based on the exponents only) since it depends on the magnitudes of the prime numbers. It is not even graded by Omega(n).

* Second rightmost diagonal: second smallest numbers of a given prime signature (A077560). (They are not increasing anymore.)

Upwards diagonals:

* Leftmost diagonal: primes. {1} (A000040)

* 2nd leftmost diagonal: squares of primes. {2} (A001248)

* 3rd leftmost diagonal: squarefree biprimes. {1,1} (A006881)

* 4th leftmost diagonal: cubes of primes. {3} (A030078)

* 5th leftmost diagonal: signature (Achilles numbers) {1,2} (A054753)

* 6th leftmost diagonal: fourth powers of primes. {4} (A030514)

* 7th leftmost diagonal: signature (Achilles numbers) {1,3} (A065036)

* 8th leftmost diagonal: squarefree triprimes. {1,1,1} (A007304)

The Achilles numbers are nonsquarefree while not perfect powers.

Prime signatures are often expressed in increasing order of exponents. The decreasing order of exponents (as on the Wiki page, see links) has the advantage of listing the exponents in the same order (with the canonical factorization convention) as the smallest number of a given prime signature.

Fifth row of A083140.

Integers k such that gcd(11*k, 210) = 1.

a(n) tells which number in array A083221 (the sieve of Eratosthenes) is at the same position where n is in Ludic array A255127. As both arrays have A005843 (even numbers) and A016945 as their two topmost rows, both sequences are among the fixed points of this permutation.

Equally: a(n) tells which number in array A083140 is at the same position where n is in the array A255129, as they are the transposes of above two arrays.