cp's OEIS Frontend

Numbers of the form p*q where p and q are primes, not necessarily distinct.

These numbers are sometimes called semiprimes or 2-almost primes.

Numbers n such that Omega(n) = 2 where Omega(n) = A001222(n) is the sum of the exponents in the prime decomposition of n.

Complement of A100959; A064911(a(n)) = 1. - Reinhard Zumkeller, Nov 22 2004

The graph of this sequence appears to be a straight line with slope 4. However, the asymptotic formula shows that the linearity is an illusion and in fact a(n)/n ~ log(n)/log(log(n)) goes to infinity. See also the graph of A066265 = number of semiprimes < 10^n.

For numbers between 33 and 15495, semiprimes are more plentiful than any other k-almost prime. See A125149.

Numbers that are divisible by exactly 2 prime powers (not including 1). - Jason Kimberley, Oct 02 2011

The (disjoint) union of A006881 and A001248. - Jason Kimberley, Nov 11 2015

An equivalent definition of this sequence is a'(n) = smallest composite number which is not divided by any smaller composite number a'(1),...,a'(n-1). - Meir-Simchah Panzer, Jun 22 2016

The above characterization can be simplified to "Composite numbers not divisible by a smaller term." This shows that this is the equivalent of primes computed via Eratosthenes's sieve, but starting with the set of composite numbers (i.e., complement of 1 union primes) instead of all positive integers > 1. It's easy to see that iterating the method (using Eratosthenes's sieve each time on the remaining numbers, complement of the previously computed set) yields numbers with bigomega = k for k = 0, 1, 2, 3, ..., i.e., {1}, A000040, this, A014612, etc. - M. F. Hasler, Apr 24 2019

For all n except n = 2, a(n) is a deficient number. - Amrit Awasthi, Sep 10 2024

It is reasonable to assume that the "comforting numbers" which John T. Williams found in Chapter 3 of Milne's book "The House at Pooh Corner" are these semiprimes. Winnie-the-Pooh wonders whether he has 14 or 15 honey pots and concludes: "It's sort of comforting." To arrange a semiprime number of honey pots in a rectangular way, let's say on a shelf, with the larger divisor parallel to the wall, there is only one solution and this is for a simple mind like Winnie-the-Pooh comforting. - Ruediger Jehn, Dec 12 2024

Also, the largest number of distinct integers such that all their pairwise differences are coprime to n. - Max Alekseyev, Mar 17 2006

The unit 1 is not a prime number (although it has been considered so in the past). 1 is the empty product of prime numbers, thus 1 has no least prime factor. - Daniel Forgues, Jul 05 2011

a(n) = least m > 0 for which n! + m and n - m are not relatively prime. - Clark Kimberling, Jul 21 2012

For n > 1, a(n) = the smallest k > 1 that divides n. - Antti Karttunen, Feb 01 2014

For n > 1, records are at prime indices. - Zak Seidov, Apr 29 2015

The initials "lpf" might be mistaken for "largest prime factor" (A009190), using "spf" for "smallest prime factor" would avoid this. - M. F. Hasler, Jul 29 2015

n = 89 is the first index > 1 for which a(n) differs from the smallest k > 1 such that (2^k + n - 2)/k is an integer. - M. F. Hasler, Aug 11 2015

From Stanislav Sykora, Jul 29 2017: (Start)

For n > 1, a(n) is also the smallest k, 1 < k <= n, for which the binomial(n,k) is not divisible by n.

Proof: (A) When k and n are relatively prime then binomial(n,k) is divisible by n because k*binomial(n,k) = n*binomial(n-1,k-1). (B) When gcd(n,k) > 1, one of its prime factors is the smallest; let us denote it p, p <= k, and consider the binomial(n,p) = (1/p!)*Product_{i=0..p-1} (n-i). Since p is a divisor of n, it cannot be a divisor of any of the remaining numerator factors. It follows that, denoting as e the largest e > 0 such that p^e|n, the numerator is divisible by p^e but not by p^(e+1). Hence, the binomial is divisible by p^(e-1) but not by p^e and therefore not divisible by n. Applying (A), (B) to all considered values of k completes the proof. (End)

From Bob Selcoe, Oct 11 2017, edited by M. F. Hasler, Nov 06 2017: (Start)

a(n) = prime(j) when n == J (mod A002110(j)), n, j >= 1, where J is the set of numbers <= A002110(j) with smallest prime factor = prime(j). The number of terms in J is A005867(j-1). So:

a(n) = 2 when n == 0 (mod 2);

a(n) = 3 when n == 3 (mod 6);

a(n) = 5 when n == 5 or 25 (mod 30);

a(n) = 7 when n == 7, 49, 77, 91, 119, 133, 161 or 203 (mod 210);

etc. (End)

For n > 1, a(n) is the leftmost term, other than 0 or 1, in the n-th row of A127093. - Davis Smith, Mar 05 2019

n-th row has length A001222(n) (n>1).

Largest nontrivial divisor of n-th semiprime. [Juri-Stepan Gerasimov, Apr 18 2010]

Greater of the prime factors of A001358(n). - Jianing Song, Aug 05 2022

Least prime factor of {n divided by the maximal power of the least prime factor of n}. - after the original name of the sequence.

a(n) = A020639(A028234(n)).

a(n) = 1 iff n is a prime power: a(A000961(n))=1 and a(A024619(n))>1.

The n chosen integers need not be distinct.

By "more than half of all integers" we mean more precisely "more than half of the integers in -m..m, for all sufficiently large m (depending on n)", and similarly with 1..m for "more than half of all positive integers".

Equivalently, a(n) is the least prime p such that more than half of all positive integers can be written as a product of primes of which n or more are not greater than p. (In this sense, a(n) might be called the median n-th least prime factor of the integers.)

The number of integers that satisfy the "product of primes" criterion for p = prime(m) is the same in every interval of primorial(m)^n integers and is A281891(m,n). Primorial(m) = A002110(m), product of the first m primes.

a(n) is the least k = prime(m) such that 2 * A281891(m,n) > A002110(m)^n.

a(n) is the least k such that more than half of all positive integers equate to the volume of an orthotope with integral sides at least n of which are orthogonal with length between 2 and k inclusive.

The next term is estimated to be a(5) ~ 3*10^18.

Square array read by descending antidiagonals: A(n,k) with rows n >= 1, columns k >= 1. 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 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 A281891.

A(n,1) = A005867(n-1) in accordance with the comment on A005867 dated Jul 16 2006.

A(2,k) = A001047(k) = 3^k - 2^k.

From Peter Munn, Jul 14 2019: (Start)

a(n) = 1 if and only if n is 1 or a prime or semiprime. Otherwise a(n) is the 3rd factor when n is written as a product of primes in nondecreasing order. For example, 60 = 2*2*3*5, so a(60) = 3.

Although values equal to 1 are predominant at low indices, their asymptotic density is 0, whereas for values equal to prime(k) for k > 0 the asymptotic density is positive, namely A281890(k,3)/A002110(k)^3. For all sufficiently large n the median value of a(1), a(2), ... a(n) is A281889(3) = 433.

(End)