cp's OEIS Frontend

Restricted growth sequence transform of triple [A010875(A020639(n)), A032742(n), A319710(n)] (with a separate value allotted for a(1)), or equally, of ordered pair [A319716(n), A319710(n)].

In addition to A319716, this filter sequence also records in the value of a(n) also the fact whether the smallest prime factor of n is unitary or not. This information is enough to determine the modulo 6 residues of all the divisors of n, thus sequences like A002324 are essentially functions of this sequence. Moreover, a lot of other information is immediately (and unavoidably) present, for example the exact prime signature of n, including also the relative order of exponents.

Any such filtering sequence can be perceived also in terms of what information it leaves out from a(n) that would be needed to reconstruct whole n from each a(n). If the whole n could be reconstructed from a(n) each time, then sequence a would be injective, and would be useless for filtering, because then it would match with any sequence. In this filter, what is left out is only the exact identity of the smallest prime factor, although its residue class mod 6 is retained. However, when the smallest prime factor is 2 or 3, this can be seen from that residue value, so for any number x in A047229, both A020639(x) and A032742(x) are known, and as x = A020639(x)*A032742(x), it means such numbers must occur in their own singleton equivalence classes.

Likewise, for any n in A283050, even if not divisible by 2 or 3, when we have A319710(n) stored in the triple as 1, this immediately gives away the exact identity of the smallest prime factor, which is equal to A014673(n) = A020639(A032742(n)) in these cases.

Thus there is a substantial subset of N (containing at least the union of A047229 and A283050) which is actually in the "blind sector" of this filter, "where anything goes", as this sequence obtains only unique values in that subdomain.

There is a related filter sequence A319996, which operates by "cleaving n from its high end" (by storing the residue class of the largest prime factor, A006530, instead of the smallest, together with n/A006530(n)), which has its own blind spots, but fortunately, they do not fully coincide with the blind spots of this filter. Naturally, any sequence like A002324 should match both to this sequence and A319996.

For all i, j:

a(i) = a(j) => A002324(i) = A002324(j),

a(i) = a(j) => A067029(i) = A067029(j),

a(i) = a(j) => A071178(i) = A071178(j),

a(i) = a(j) => A077462(i) = A077462(j) => A101296(i) = A101296(j),

a(i) = a(j) => A319716(i) = A319716(j) => A319690(i) = A319690(j).

Numbers n = A020639(n) * A014673(n) * A054576(n), for which A020639(n)^2 < A014673(n) and either A054576(n) = 1 or A032742(n) satisfies the same condition (is the term of this sequence).

Square array read by descending antidiagonals; A(n,k) with rows n >= 0, columns k >= 0. Prime factors are counted with multiplicity. Primorial(n) = A002110(n): product of first n primes.

Visualize the prime factorizations of the positive integers as a table with row headings giving each successive integer, and the primes of which the row heading is the product listed across the columns in nondecreasing order, repeated when necessary. Except for 1, which lacks prime factors, column 1 has the row heading's least prime factor, column 2 has a value for composite numbers but is blank for primes, and so on. This sequence measures precisely how frequently values up to and including the various primes occur in each column. This is possible because any given prime occurs cyclically in any given column, for the reason following.

The occurrence pattern of up to k factors of prime(n) in such prime factorizations has a fundamental period over the positive integers of prime(n)^k. The least common period for up to k factors of each of the first n primes is primorial(n)^k, and this covers everything that can affect the occurrence of prime(n) in the least k factors. Thus prime(n) is k-th least prime factor of integer m if and only if it is k-th least prime factor of m + primorial(n)^k.

Intermediate values in the calculation of this sequence appear in A281890.

If n > 0, A(n,1) = A053144(n) in accordance with the comment on A053144 dated Apr 08 2010.

A(2,k) = A027649(k) = 2*(3^k) - 2^k.

After the initial a(0)=1, the third row of array A285321 divided by its first row. After 1, all terms are either primes or squares of primes. See A285110.

The sequence is completely determined by the positions of two least significant 1-bits of n: After initial zero, if n is a power of two (only one 1-bit present) or if prime(1+A285099(n)) > prime(1+A007814(n))^2, a(n) = prime(1+A007814(n))^2 = A020639(A019565(n))^2, otherwise a(n) = prime(1+A285099(n)) = A014673(A019565(n)).

Numbers divisible by at least one of 4, 6, 9, 10, 14, 15, 21, 25, 35, 49.

Exactly half of the first 10, first 100 and first 600 positive integers are divisible by a 7-smooth composite number; the largest 7-smooth divisor of the remaining numbers is 1, 2, 3, 5 or 7.

Intervals extending to hundreds of integers with exactly 50% membership of this sequence are far from rare, some notable examples being [3000, 3999], [8000, 8999], [20000, 20999], [21000, 21999] and [23000, 23999]. This reflects the asymptotic density of the corresponding set being close to 0.5, precisely 1847 / 3675 = 0.50258503... (and membership of the set has a periodic pattern). See A343598 for further information.