cp's OEIS Frontend

From Antti Karttunen, Dec 06 2014: (Start)

For n >= 2, a(n) tells in which column of the sieve of Eratosthenes (see A083140, A083221) n occurs in. A055396 gives the corresponding row index.

(End)

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

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 n > 1 for which there exists r <= gpf(n) such that r^k <= lpf(n) and gpf(n) < r^(k+1) for some k >= 0, where lpf and gpf (least and greatest prime factor of n) are given by A020639(n) and A006530(n) (the original, equivalent definition of the sequence).

Numbers n > 1 such that A252375(n) < 1 + A006530(n). Equally, one can substitute A251725 for A252375.

These are numbers n all of whose prime factors "fit between" two consecutive powers of some positive integer which itself is <= the largest prime factor of n.

Conjecture: If any n is in the sequence, then so is A003961(n).

Note: if Legendre's or Brocard's conjecture is true, then the above conjecture is true as well. See my comments at A251728. - Antti Karttunen, Jan 01 2015

a(n) = 1 iff A001358(n) is the square of a prime (A001248);

a(A174956(A138511(n))) = A084127(A174956(A138511(n))) + 1.

Equally, 1 if A001358(n) = p^2, otherwise, if A001358(n) = p*q (p, q primes, p < q), then a(n) = A252375(n) = the least r such that r^k <= p < q < r^(k+1), for some k >= 0. - Antti Karttunen, Dec 16 2014

a(A174956(A085721(n))) <= 2. - Reinhard Zumkeller, Dec 19 2014

This sequence was suggested by Moshe Shmuel Newman.

A020639(n)^2 < a(n) < A020639(n)^3. - Reinhard Zumkeller, Dec 17 2014

In other words, k = p*q with primes p, q satisfying p < q < p^2. - Charles R Greathouse IV, Apr 03 2017

If "strictly less than" in the definition were changed to "less than or equal to" then this sequence would also include the squares of primes (A001248), resulting in A251728. - Jon E. Schoenfield, Dec 27 2022

In other words, {1} together with primes and such composite numbers n = p_i * p_j * p_k * ... * p_u, p_i <= p_j <= p_k <= ... <= p_u, where each successive prime factor (when sorted into a nondecreasing order) is less than the square of the previous: (p_i)^2 > p_j, (p_j)^2 > p_k, etc.

Whenever gcd(a(i),a(j)) > 1, then a(i)*a(j) and lcm(a(i),a(j)) are also members of this sequence.

For n >= 2: a(n) = the first "settled semiprime" in the column n of the sieve of Eratosthenes: a(n) = A083221(A251719(n), n).

The "settling of semiprimes" here means that from that semiprime onward, all the other terms in the same column n of a square array A083221 (which is constructed from the sieve of Eratosthenes) are also semiprimes, obtained by successive iterations of A003961 starting from the semiprime here given as a(n). Cf. comments in A251728 which contains all such semiprimes. The "unsettled" semiprimes are in its complement A138511.

Here we assume 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).

Positions of zeros in A280692. Conjectured to be also the positions of ones in A280703.

For most terms k of this sequence A003961(k) is also included as a term. Exceptions are: 50, 52, 66, 186, 435, 1245, 1445, 2068, 2085, 11605, ... that seems to be a subsequence of all those terms that have more than two prime factors: 50, 52, 66, 186, 435, 1245, 1445, 2068, 2085, 8695, 11605, ...

Note how 8695 = 5*37*47 and A003961(8695) = 7*41*53 = 15211 = A003961(8695) = A250469(8695) (for no apparent reason?).